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</button> <ul id="toc-Voting_and_counting-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Point_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Point_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Point distributions</span> </div> </a> <button aria-controls="toc-Point_distributions-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 Point distributions subsection</span> </button> <ul id="toc-Point_distributions-sublist" class="vector-toc-list"> <li 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href="#Comparison_of_progression_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Comparison of progression types</span> </div> </a> <ul id="toc-Comparison_of_progression_types-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analysis_of_non-ranking_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analysis_of_non-ranking_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Analysis of non-ranking systems</span> </div> </a> <ul id="toc-Analysis_of_non-ranking_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparative_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparative_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Comparative examples</span> </div> </a> <ul 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class="vector-toc-numb">5.1</span> <span>IIA example</span> </div> </a> <ul id="toc-IIA_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-IoC_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#IoC_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>IoC example</span> </div> </a> <ul id="toc-IoC_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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 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role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/Positional_voting" title="Special:EditPage/Positional voting">improve it</a></b> or discuss these issues on the <b><a href="/wiki/Talk:Positional_voting" title="Talk:Positional voting">talk page</a></b>. <small><i>(<a 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Positional+voting%22">"Positional voting"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Positional+voting%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Positional+voting%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Positional+voting%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Positional+voting%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Positional+voting%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-One_source plainlinks metadata ambox ambox-content ambox-one_source" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies largely or entirely on a <a href="/wiki/Wikipedia:Articles_with_a_single_source" title="Wikipedia:Articles with a single source">single source</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Positional_voting##" title="Talk:Positional voting">talk page</a>. 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.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 class="mw-selflink selflink">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 noviewer" typeof="mw:File"><a href="/wiki/File:A_coloured_voting_box.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/16px-A_coloured_voting_box.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/24px-A_coloured_voting_box.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/32px-A_coloured_voting_box.svg.png 2x" data-file-width="160" data-file-height="160" /></a></span> </span><a href="/wiki/Portal:Politics" title="Portal:Politics">Politics portal</a></li><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Emblem-money.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/16px-Emblem-money.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/24px-Emblem-money.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/32px-Emblem-money.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span> </span><a href="/wiki/Portal:Economics" title="Portal:Economics">Economics portal</a></li></ul></div><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electoral_systems_sidebar" title="Template:Electoral systems sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electoral_systems_sidebar" title="Template talk:Electoral systems sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electoral_systems_sidebar" title="Special:EditPage/Template:Electoral systems sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Positional voting</b> is a <a href="/wiki/Ranked_voting" title="Ranked voting">ranked voting</a> <a href="/wiki/Electoral_system" title="Electoral system">electoral system</a> in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins.<sup id="cite_ref-Saari_1-0" class="reference"><a href="#cite_note-Saari-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will (<a href="/wiki/Voting_at_the_Eurovision_Song_Contest" title="Voting at the Eurovision Song Contest">Eurovision Song Contest</a>) or it may form a mathematical sequence such as an arithmetic progression (<a href="/wiki/Borda_count" title="Borda count">Borda count</a>), a geometric one (<a href="/wiki/Positional_notation" title="Positional notation">positional number system</a>) or a harmonic one (<a href="/wiki/Borda_count#Dowdall_system_(Nauru)" title="Borda count">Nauru/Dowdall method</a>). The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes. </p><p>Positional voting should be distinguished from <a href="/wiki/Score_voting" title="Score voting">score voting</a>: in the former, the score that each voter gives to each candidate is uniquely determined by the candidate's rank; in the latter, each voter is free to give any score to any candidate. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Voting_and_counting">Voting and counting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=1" title="Edit section: Voting and counting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In positional voting, voters complete a <a href="/wiki/Ranked_ballot" class="mw-redirect" title="Ranked ballot">ranked ballot</a> by expressing their preferences in rank order. The rank position of each voter preference is allotted a specific fixed weighting. Typically, the higher the rank of the preference, the more points it is worth. Occasionally, it may share the same weighting as a lower-ranked preference but it is never worth fewer points. </p><p>Usually, every voter is required to express a unique <a href="/wiki/Ordinal_numeral" title="Ordinal numeral">ordinal</a> preference for each option on the ballot in strict descending rank order. However, a particular positional voting system may permit voters to truncate their preferences after expressing one or more of them and to leave the remaining options unranked and consequently worthless. Similarly, some other systems may limit the number of preferences that can be expressed. For example, in the <a href="/wiki/Eurovision_Song_Contest" title="Eurovision Song Contest">Eurovision Song Contest</a> only their top ten preferences are ranked by each country although many more than ten songs compete in the contest. Again, unranked preferences have no value. In positional voting, ranked ballots with tied options are normally considered as invalid. </p><p>The counting process is straightforward. All the preferences cast by voters are awarded the points associated with their rank position. Then, all the points for each option are tallied and the one with the most points is the winner. Where a few winners (<span class="texhtml mvar" style="font-style:italic;">W</span>) are instead required following the count, the <span class="texhtml mvar" style="font-style:italic;">W</span> highest-ranked options are selected. Positional voting is not only a means of identifying a single winner but also a method for converting sets of individual preferences (ranked ballots) into one collective and fully rank-ordered set. It is possible and legitimate for options to be tied in this resultant set; even in first place. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a positional voting election for choosing a single winner from three options A, B and C. No truncation or ties are permitted and a first, second and third preference is here worth 4, 2 and 1 point respectively. There are then six different ways in which each voter may rank order these options. The 100 voters cast their ranked ballots as follows: </p> <table class="wikitable"> <tbody><tr> <th>Number of ballots</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>24 </th> <td>A</td> <td>B</td> <td>C </td></tr> <tr> <th>18 </th> <td>A</td> <td>C</td> <td>B </td></tr> <tr> <th>12 </th> <td>B</td> <td>A</td> <td>C </td></tr> <tr> <th>16 </th> <td>B</td> <td>C</td> <td>A </td></tr> <tr> <th>20 </th> <td>C</td> <td>A</td> <td>B </td></tr> <tr> <th>10 </th> <td>C</td> <td>B</td> <td>A </td></tr></tbody></table> <p>After voting closes, the points awarded by the voters are then tallied and the options ranked according to the points total. </p> <table class="wikitable"> <tbody><tr> <th>Option</th> <th>Points to be tallied</th> <th>Total</th> <th>Overall rank </th></tr> <tr> <th>A </th> <td>(24 + 18) x 4 + (12 + 20) x 2 + (16 + 10) x 1</td> <td>258</td> <td>First </td></tr> <tr> <th>B </th> <td>(12 + 16) x 4 + (24 + 10) x 2 + (18 + 20) x 1</td> <td>218</td> <td>Third </td></tr> <tr> <th>C </th> <td>(20 + 10) x 4 + (18 + 16) x 2 + (24 + 12) x 1</td> <td>224</td> <td>Second </td></tr></tbody></table> <p>Therefore, having the highest tally, option A is the winner here. Note that the election result also generates a full ranking of all the options. </p> <div class="mw-heading mw-heading2"><h2 id="Point_distributions">Point distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=3" title="Edit section: Point distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For positional voting, any distribution of points to the rank positions is valid, so long as the points are <a href="/wiki/Weakly_decreasing" class="mw-redirect" title="Weakly decreasing">weakly decreasing</a> in the rank of each candidate. In other words, a worse-ranked candidate must receive fewer points than a better-ranked candidate.<sup id="cite_ref-Saari_1-1" class="reference"><a href="#cite_note-Saari-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Borda_(Unbiased)"><span id="Borda_.28Unbiased.29"></span>Borda (Unbiased)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=4" title="Edit section: Borda (Unbiased)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classic example of a positional voting electoral system is the <a href="/wiki/Borda_count" title="Borda count">Borda count</a>.<sup id="cite_ref-Saari_1-2" class="reference"><a href="#cite_note-Saari-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Typically, for a single-winner election with <span class="texhtml mvar" style="font-style:italic;">N</span> candidates, a first preference is worth <span class="texhtml mvar" style="font-style:italic;">N</span> points, a second preference <span class="texhtml"><i>N</i> – 1</span> points, a third preference <span class="texhtml"><i>N</i> – 2</span> points and so on until the last (<span class="texhtml mvar" style="font-style:italic;">N</span>th) preference that is worth just 1 point. So, for example, the points are respectively 4, 3, 2 and 1 for a four-candidate election. </p><p>Mathematically, the point value or weighting (<span class="texhtml mvar" style="font-style:italic;">w<sub>n</sub></span>) associated with a given rank position (<span class="texhtml mvar" style="font-style:italic;">n</span>) is defined below; where the weighting of the first preference is <span class="texhtml mvar" style="font-style:italic;">a</span> and the common difference is <span class="texhtml mvar" style="font-style:italic;">d</span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n}=a-(n-1)d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{n}=a-(n-1)d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7bebf8cc11e4786ae933e11468d72c3093c30a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.474ex; height:2.843ex;" alt="{\displaystyle w_{n}=a-(n-1)d}"></span> </p><p>where <span class="texhtml"><i>a</i> = <i>N</i></span>, the number of candidates. </p><p>The value of the first preference need not be <span class="texhtml mvar" style="font-style:italic;">N</span>. It is sometimes set to <span class="texhtml"><i>N</i> – 1</span> so that the last preference is worth zero. Although it is convenient for counting, the common difference need not be fixed at one since the overall ranking of the candidates is unaffected by its specific value. Hence, despite generating differing tallies, any value of <span class="texhtml mvar" style="font-style:italic;">a</span> or <span class="texhtml mvar" style="font-style:italic;">d</span> for a Borda count election will result in identical candidate rankings.<sup id="cite_ref-Saari_1-3" class="reference"><a href="#cite_note-Saari-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The consecutive Borda count weightings form an <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Top-heavy">Top-heavy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=5" title="Edit section: Top-heavy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Common systems for evaluating preferences, other than Borda, are typically "top-heavy". In other words, the method focuses on how many voters consider a candidate one of their "favourites". </p> <div class="mw-heading mw-heading4"><h4 id="Plurality_voting">Plurality voting</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=6" title="Edit section: Plurality voting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/First-preference_plurality" class="mw-redirect" title="First-preference plurality">First-preference plurality</a></div> <p>Under <a href="/wiki/Plurality_voting" title="Plurality voting">first-preference plurality</a> (FPP), the most-preferred option receives 1 point while all other options receive 0 points each. This is the most top-heavy positional voting system. </p> <div class="mw-heading mw-heading4"><h4 id="Geometric">Geometric</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=7" title="Edit section: Geometric"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An alternative mathematical <a href="/wiki/Sequence" title="Sequence">sequence</a> known as a <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a> may also be used in positional voting. Here, there is instead a common ratio <span class="texhtml mvar" style="font-style:italic;">r</span> between adjacent weightings. In order to satisfy the two validity conditions, the value of <span class="texhtml mvar" style="font-style:italic;">r</span> must be less than one so that weightings decrease as preferences descend in rank. Where the value of the first preference is <span class="texhtml mvar" style="font-style:italic;">a</span>, the weighting (<span class="texhtml mvar" style="font-style:italic;">w<sub>n</sub></span>) awarded to a given rank position (<span class="texhtml mvar" style="font-style:italic;">n</span>) is defined below. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n}=ar^{n-1},\qquad 0\leq r<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{n}=ar^{n-1},\qquad 0\leq r<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981e6d2a9aaeb60c5df1f56e4280276110d97b39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.828ex; height:3.009ex;" alt="{\displaystyle w_{n}=ar^{n-1},\qquad 0\leq r<1}"></span> </p><p>For example, the sequence of consecutively halved weightings of 1, 1/2, 1/4, 1/8, … as used in the <a href="/wiki/Binary_number" title="Binary number">binary number</a> system constitutes a geometric progression with a common ratio of one-half (<span class="texhtml mvar" style="font-style:italic;">r</span> = 1/2). Such weightings are inherently valid for use in positional voting systems provided that a legitimate common ratio is employed. Using a common ratio of zero, this form of positional voting has weightings of 1, 0, 0, 0, … and so produces ranking outcomes identical to that for first-past-the-post or <a href="/wiki/Plurality_voting" title="Plurality voting">plurality voting</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Dowdall_system_(Nauru)"><span id="Dowdall_system_.28Nauru.29"></span><span class="anchor" id="Dowdall_system"></span><span class="anchor" id="Dowdall_method"></span><span class="anchor" id="Dowdall's_method"></span><span class="anchor" id="Dowdall"></span>Dowdall system (Nauru)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=8" title="Edit section: Dowdall system (Nauru)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternatively, the denominators of the above fractional weightings could form an arithmetic progression instead; namely 1/1, 1/2, 1/3, 1/4 and so on down to <span class="texhtml">1/<i>N</i></span>. This further mathematical sequence is an example of a <a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">harmonic progression</a>. These particular descending rank-order weightings are in fact used in <span class="texhtml mvar" style="font-style:italic;">N</span>-candidate positional voting elections to the <a href="/wiki/Elections_in_Nauru" title="Elections in Nauru">Nauru parliament</a>.<sup id="cite_ref-Nauru_2019_2-0" class="reference"><a href="#cite_note-Nauru_2019-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nauru_2022_3-0" class="reference"><a href="#cite_note-Nauru_2022-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For such electoral systems, the weighting (<span class="texhtml mvar" style="font-style:italic;">w<sub>n</sub></span>) allocated to a given rank position (<span class="texhtml mvar" style="font-style:italic;">n</span>) is defined below; where the value of the first preference is <span class="texhtml mvar" style="font-style:italic;">a</span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n}={\frac {a^{2}}{a+(n-1)d}}={\frac {a}{1+{\frac {(n-1)d}{a}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>d</mi> </mrow> <mi>a</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{n}={\frac {a^{2}}{a+(n-1)d}}={\frac {a}{1+{\frac {(n-1)d}{a}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b316d71be68118ef91860ad7d0486787fa253557" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:33.956ex; height:7.676ex;" alt="{\displaystyle w_{n}={\frac {a^{2}}{a+(n-1)d}}={\frac {a}{1+{\frac {(n-1)d}{a}}}},}"></span> </p><p>where <span class="texhtml"><i>w</i><sub>1</sub> = <i>a</i></span>. </p><p>For the Nauru system, the first preference <span class="texhtml mvar" style="font-style:italic;">a</span> is worth one and the common difference <span class="texhtml mvar" style="font-style:italic;">d</span> between adjacent denominators is also one. Numerous other harmonic sequences can also be used in positional voting. For example, setting <span class="texhtml mvar" style="font-style:italic;">a</span> to 1 and <span class="texhtml mvar" style="font-style:italic;">d</span> to 2 generates the reciprocals of all the odd numbers (1, 1/3, 1/5, 1/7, …) whereas letting <span class="texhtml mvar" style="font-style:italic;">a</span> be 1/2 and <span class="texhtml mvar" style="font-style:italic;">d</span> be 1/2 produces those of all the even numbers (1/2, 1/4, 1/6, 1/8, …). </p><p>The harmonic variant used by the island nation of <a href="/wiki/Nauru" title="Nauru">Nauru</a> is called the <a href="/wiki/Borda_count#Dowdall_system_(Nauru)" title="Borda count">Dowdall</a> system as it was devised by Nauru's Secretary for Justice (Desmond Dowdall) in 1971.<sup id="cite_ref-Reilly20022_4-0" class="reference"><a href="#cite_note-Reilly20022-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Fraenkel_Grofman_20142_5-0" class="reference"><a href="#cite_note-Fraenkel_Grofman_20142-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Here, each voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives <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> a point, the 3rd-ranked candidate receives <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span> of a point, etc. When counting candidate tallies in Nauru, decimal numbers rounded to three places after the decimal point are employed rather than fractions. (This system should not be confused with the use of <a href="/wiki/Highest_averages_method" title="Highest averages method">sequential divisors</a> in proportional systems such as <a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">proportional approval voting</a>, an unrelated method.) A similar system of weighting lower-preference votes was used in the 1925 <a href="/wiki/Oklahoma_primary_electoral_system" title="Oklahoma primary electoral system">Oklahoma primary electoral system</a>. </p><p>For a four-candidate election, the Dowdall point distribution would be this: </p> <table class="wikitable"> <tbody><tr> <th>Ranking </th> <th>Candidate </th> <th>Formula </th> <th>Points </th></tr> <tr> <th>1st </th> <td>Andrew </td> <td>1/1 </td> <td><b>1.000</b> </td></tr> <tr> <th>2nd </th> <td>Brian </td> <td>1/2 </td> <td><b>0.500</b> </td></tr> <tr> <th>3rd </th> <td>Catherine </td> <td>1/3 </td> <td><b>0.333</b> </td></tr> <tr> <th>4th </th> <td>David </td> <td>1/4 </td> <td><b>0.250</b> </td></tr></tbody></table> <p>This method is more favourable to candidates with many first preferences than the conventional Borda count. It has been described as a system "somewhere between plurality and the Borda count, but as veering more towards plurality".<sup id="cite_ref-Fraenkel_Grofman_20142_5-1" class="reference"><a href="#cite_note-Fraenkel_Grofman_20142-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules.<sup id="cite_ref-Fraenkel_Grofman_20142_5-2" class="reference"><a href="#cite_note-Fraenkel_Grofman_20142-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Eurovision">Eurovision</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=9" title="Edit section: Eurovision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Eurovision_Song_Contest" title="Eurovision Song Contest">Eurovision Song Contest</a> uses a first preference worth 12 points, while a second one is given 10 points. The next eight consecutive preferences are awarded 8, 7, 6, 5, 4, 3, 2 and 1 point. All remaining preferences receive zero points. </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_of_progression_types">Comparison of progression types</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=10" title="Edit section: Comparison of progression types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In positional voting, the weightings (<span class="texhtml mvar" style="font-style:italic;">w</span>) of consecutive preferences from first to last decline monotonically with rank position (<span class="texhtml mvar" style="font-style:italic;">n</span>). However, the rate of decline varies according to the type of progression employed. Lower preferences are more influential in election outcomes where the chosen progression employs a sequence of weightings that descend relatively slowly with rank position. The more slowly weightings decline, the more consensual and less polarising positional voting becomes. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Positional_Voting_Progressions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Positional_Voting_Progressions.png/220px-Positional_Voting_Progressions.png" decoding="async" width="220" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Positional_Voting_Progressions.png/330px-Positional_Voting_Progressions.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Positional_Voting_Progressions.png/440px-Positional_Voting_Progressions.png 2x" data-file-width="700" data-file-height="600" /></a><figcaption>Relative decline in preference weightings with descending rank order for four positional voting electoral systems</figcaption></figure> <p>This figure illustrates such declines over ten preferences for the following four positional voting electoral systems: </p> <ul><li>Borda count (where <span class="texhtml"><i>a</i> = <i>N</i> = 10</span> and <span class="texhtml"><i>d</i> = 1</span>)</li> <li>Binary number system (where <span class="texhtml"><i>a</i> = 1</span> and <span class="texhtml"><i>r</i> = 1/2</span>)</li> <li>Nauru method (where <span class="texhtml"><i>a</i> = 1</span> and <span class="texhtml"><i>d</i> = 1</span>)</li> <li>Eurovision Song Contest (non-zero preferences only)</li></ul> <p>To aid comparison, the actual weightings have been normalised; namely that the first preference is set at one and the other weightings in the particular sequence are scaled by the same factor of <span class="texhtml">1/<i>a</i></span>. </p><p>The relative decline of weightings in any arithmetic progression is constant as it is not a function of the common difference <span class="texhtml mvar" style="font-style:italic;">d</span>. In other words, the relative difference between adjacent weightings is fixed at <span class="texhtml">1/<i>N</i></span>. In contrast, the value of <span class="texhtml mvar" style="font-style:italic;">d</span> in a harmonic progression does affect the rate of its decline. The higher its value, the faster the weightings descend. Whereas the lower the value of the common ratio <span class="texhtml mvar" style="font-style:italic;">r</span> for a geometric progression, the faster its weightings decline. </p><p>The weightings of the digit positions in the binary number system were chosen here to highlight an example of a geometric progression in positional voting. In fact, the consecutive weightings of any <a href="/wiki/Radix" title="Radix">digital number system</a> can be employed since they all constitute geometric progressions. For example, the binary, ternary, octal and decimal number systems use a <a href="/wiki/Radix" title="Radix">radix</a> <span class="texhtml mvar" style="font-style:italic;">R</span> of 2, 3, 8 and 10 respectively. The value <span class="texhtml mvar" style="font-style:italic;">R</span> is also the common ratio of the geometric progression going up in rank order while <span class="texhtml mvar" style="font-style:italic;">r</span> is the complementary common ratio descending in rank. Therefore, <span class="texhtml mvar" style="font-style:italic;">r</span> is the reciprocal of <span class="texhtml mvar" style="font-style:italic;">R</span> and the <span class="texhtml mvar" style="font-style:italic;">r</span> ratios are respectively 1/2, 1/3, 1/8 and 1/10 for these positional number systems when employed in positional voting. </p><p>As it has the smallest radix, the rate of decline in preference weightings is slowest when using the binary number system. Although the radix <span class="texhtml mvar" style="font-style:italic;">R</span> (the number of unique digits used in the number system) has to be an integer, the common ratio <span class="texhtml mvar" style="font-style:italic;">r</span> for positional voting does not have to be the reciprocal of such an integer. Any value between zero and just less than one is valid. For a slower descent of weightings than that generated using the binary number system, a common ratio greater than one-half must be employed. The higher the value of <span class="texhtml mvar" style="font-style:italic;">r</span>, the slower the decrease in weightings with descending rank. </p> <div class="mw-heading mw-heading2"><h2 id="Analysis_of_non-ranking_systems">Analysis of non-ranking systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=11" title="Edit section: Analysis of non-ranking systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although not categorised as positional voting electoral systems, some non-ranking methods can nevertheless be analysed mathematically as if they were by allocating points appropriately.<sup id="cite_ref-Saari_1-4" class="reference"><a href="#cite_note-Saari-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Given the absence of strict monotonic ranking here, all favoured options are weighted identically with a high value and all the remaining options with a common lower value. The two validity criteria for a sequence of weightings are hence satisfied. </p><p>For an <span class="texhtml mvar" style="font-style:italic;">N</span>-candidate ranked ballot, let the permitted number of favoured candidates per ballot be <span class="texhtml mvar" style="font-style:italic;">F</span> and the two weightings be one point for these favoured candidates and zero points for those not favoured. When analytically represented using positional voting, favoured candidates must be listed in the top <span class="texhtml mvar" style="font-style:italic;">F</span> rank positions in any order on each ranked ballot and the other candidates in the bottom <span class="texhtml"><i>N</i>-<i>F</i></span> rank positions. This is essential as the weighting of each rank position is fixed and common to each and every ballot in positional voting. </p><p>Unranked single-winner methods that can be analysed as positional voting electoral systems include: </p> <ul><li><a href="/wiki/Plurality_voting" title="Plurality voting">Plurality voting</a> (FPTP): The most preferred option receives 1 point; all other options receive 0 points each. (<span class="texhtml"><i>F</i> = 1</span>)</li> <li><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Anti-plurality voting</a>: The least preferred option receives 0 points; all other options receive 1 point each. (<span class="texhtml"><i>F</i> = <i>N</i> – 1</span>)</li></ul> <p>And unranked methods for multiple-winner elections (with <span class="texhtml mvar" style="font-style:italic;">W</span> winners) include: </p> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">Single non-transferable vote</a>: The most preferred option receives 1 point; all other options receive 0 points each. (<span class="texhtml"><i>F</i> = 1</span>)</li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a>: The <span class="texhtml mvar" style="font-style:italic;">X</span> most preferred options (where <span class="texhtml">1 < <i>X</i> < <i>W</i></span>) receive 1 point each; all other options receive 0 points each. (<span class="texhtml"><i>F</i> = <i>X</i></span>)</li> <li><a href="/wiki/Plurality-at-large_voting" class="mw-redirect" title="Plurality-at-large voting">Bloc voting</a>: The <span class="texhtml mvar" style="font-style:italic;">W</span> most preferred options receive 1 point each; all other options receive 0 points each. (<span class="texhtml"><i>F</i> = <i>W</i></span>)</li></ul> <p>In <a href="/wiki/Approval_voting" title="Approval voting">approval voting</a>, voters are free to favour as many or as few candidates as they wish so <span class="texhtml mvar" style="font-style:italic;">F</span> is not fixed but varies according to the individual ranked ballots being cast. As rank positions would then have different weightings on different ballots, approval voting is not a positional voting system; nor can it be analysed as such. </p> <div class="mw-heading mw-heading2"><h2 id="Comparative_examples">Comparative examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=12" title="Edit section: Comparative examples"><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>Where <span class="texhtml mvar" style="font-style:italic;">w<sub>n</sub></span> is the weighting of the <span class="texhtml mvar" style="font-style:italic;">n</span>th preference, the following table defines the resultant tally calculation for each city: </p> <table class="wikitable"> <tbody><tr> <th>Voters' home city</th> <th>Vote tally per 1200 voters </th></tr> <tr> <th>Memphis </th> <td>(42w<sub>1</sub> + 26w<sub>4</sub> + 15w<sub>4</sub> + 17w<sub>4</sub>) x 1200/100 </td></tr> <tr> <th>Nashville </th> <td>(42w<sub>2</sub> + 26w<sub>1</sub> + 15w<sub>3</sub> + 17w<sub>3</sub>) x 1200/100 </td></tr> <tr> <th>Chattanooga </th> <td>(42w<sub>3</sub> + 26w<sub>2</sub> + 15w<sub>1</sub> + 17w<sub>2</sub>) x 1200/100 </td></tr> <tr> <th>Knoxville </th> <td>(42w<sub>4</sub> + 26w<sub>3</sub> + 15w<sub>2</sub> + 17w<sub>1</sub>) x 1200/100 </td></tr></tbody></table> <p>For a first preference worth <span class="texhtml"><i>w</i><sub>1</sub> = 1</span>, the table below states the value of each of the four weightings for a range of different positional voting systems that could be employed for this election: </p> <table class="wikitable"> <tbody><tr> <th>Voting system</th> <th><span class="texhtml"><i>w</i><sub>1</sub></span></th> <th><span class="texhtml"><i>w</i><sub>2</sub></span></th> <th><span class="texhtml"><i>w</i><sub>3</sub></span></th> <th><span class="texhtml"><i>w</i><sub>4</sub></span></th> <th>Sum </th></tr> <tr> <th>Plurality </th> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td>1 </td></tr> <tr> <th>Binary number system </th> <td>1</td> <td>1/2</td> <td>1/4</td> <td>1/8</td> <td>1.875 </td></tr> <tr> <th>Nauru method </th> <td>1</td> <td>1/2</td> <td>1/3</td> <td>1/4</td> <td>2.083 </td></tr> <tr> <th>Borda count </th> <td>1</td> <td>3/4</td> <td>1/2</td> <td>1/4</td> <td>2.5 </td></tr> <tr> <th>Anti-plurality </th> <td>1</td> <td>1</td> <td>1</td> <td>0</td> <td>3 </td></tr></tbody></table> <p>These five positional voting systems are listed in <a class="mw-selflink-fragment" href="#Comparison_of_progression_types">progression type</a> order. The slower the decline in weighting values with descending rank order, the greater is the sum of the four weightings; see end column. Plurality declines the fastest while anti-plurality is the slowest. </p><p>For each positional voting system, the tallies for each of the four city options are determined from the above two tables and stated below: </p> <table class="wikitable"> <tbody><tr> <th>Voting system</th> <th>Memphis</th> <th>Nashville</th> <th>Chattanooga</th> <th>Knoxville </th></tr> <tr> <th>Plurality </th> <td>504</td> <td>312</td> <td>180</td> <td>204 </td></tr> <tr> <th>Binary number system </th> <td>591</td> <td>660</td> <td>564</td> <td>435 </td></tr> <tr> <th>Nauru method </th> <td>678</td> <td>692</td> <td>606</td> <td>524 </td></tr> <tr> <th>Borda count </th> <td>678</td> <td>882</td> <td>819</td> <td>621 </td></tr> <tr> <th>Anti-plurality </th> <td>504</td> <td>1200</td> <td>1200</td> <td>696 </td></tr></tbody></table> <p>For each potential positional voting system that could be used in this election, the consequent overall rank order of the options is shown below: </p> <table class="wikitable"> <tbody><tr> <th>Voting system</th> <th>First place</th> <th>Second place</th> <th>Third place</th> <th>Fourth place </th></tr> <tr> <th>Plurality </th> <td>Memphis</td> <td>Nashville</td> <td>Knoxville</td> <td>Chattanooga </td></tr> <tr> <th>Binary number system </th> <td>Nashville</td> <td>Memphis</td> <td>Chattanooga</td> <td>Knoxville </td></tr> <tr> <th>Nauru method </th> <td>Nashville</td> <td>Memphis</td> <td>Chattanooga</td> <td>Knoxville </td></tr> <tr> <th>Borda count </th> <td>Nashville</td> <td>Chattanooga</td> <td>Memphis</td> <td>Knoxville </td></tr> <tr> <th>Anti-plurality </th> <td>Chattanooga / Nashville</td> <td></td> <td>Knoxville</td> <td>Memphis </td></tr></tbody></table> <p>This table highlights the importance of <a class="mw-selflink-fragment" href="#Comparison_of_progression_types">progression type</a> in determining the winning outcome. With all voters either strongly for or against Memphis, it is a very ‘polarized’ option so Memphis finishes first under plurality and last with anti-plurality. Given its central location, Nashville is the ‘consensus’ option here. It wins under the Borda count and the two other non-polarized systems </p> <div class="mw-heading mw-heading2"><h2 id="Evaluation_against_voting_system_criteria">Evaluation against voting system criteria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=13" title="Edit section: Evaluation against voting system criteria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a class of voting systems, positional voting can be evaluated against objective <a href="/wiki/Comparison_of_electoral_systems#Mathematical_criteria" class="mw-redirect" title="Comparison of electoral systems">mathematical criteria</a> to evaluate its strengths and weaknesses in comparison with other single-winner electoral methods. </p><p>Positional voting satisfies the following criteria: </p> <ul><li><a href="/wiki/Non-dictatorship" class="mw-redirect" title="Non-dictatorship">Non-dictatorship</a></li> <li><a href="/wiki/Unrestricted_domain" title="Unrestricted domain">Unrestricted domain</a></li> <li><a href="/wiki/Comparison_of_electoral_systems#Mathematical_criteria" class="mw-redirect" title="Comparison of electoral systems">Summability (with order N)</a></li> <li><a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">Consistency</a></li> <li><a href="/wiki/Participation_criterion" class="mw-redirect" title="Participation criterion">Participation</a></li> <li><a href="/wiki/Resolvability_criterion" title="Resolvability criterion">Resolvability</a></li> <li><a href="/wiki/Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">Monotonicity</a></li> <li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li></ul> <p>But it fails to satisfy the following criteria: </p> <ul><li><a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">Independence of Irrelevant Alternatives</a> (IIA)</li> <li><a href="/wiki/Independence_of_clones_criterion" title="Independence of clones criterion">Independence of Clones</a> (IoC)</li> <li><a href="/wiki/Condorcet_criterion" class="mw-redirect" title="Condorcet criterion">Condorcet winner</a></li> <li><a href="/wiki/Condorcet_loser_criterion" title="Condorcet loser criterion">Condorcet loser</a> (except the Borda count)</li> <li><a href="/wiki/Reversal_symmetry" class="mw-redirect" title="Reversal symmetry">Reversal symmetry</a> (except the Borda count)</li> <li><a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">Majority</a> (except when equivalent to plurality)</li></ul> <p>According to <a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow’s impossibility theorem</a>, no ranked voting system can satisfy all of the following four criteria when collectively ranking three or more alternatives: </p> <ul><li><a href="/wiki/Non-dictatorship" class="mw-redirect" title="Non-dictatorship">Non-dictatorship</a></li> <li><a href="/wiki/Unrestricted_domain" title="Unrestricted domain">Unrestricted domain</a></li> <li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li> <li><a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">Independence of Irrelevant Alternatives</a> (IIA)</li></ul> <p>Prior to voter preferences being cast, voting systems that treat all voters as equals and all candidates as equals pass the first two criteria above. So, like any other ranking system, positional voting cannot pass both of the other two. It is <a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficient</a> but is not <a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">independent of irrelevant alternatives</a>. This failure means that the addition or deletion of a non-winning (irrelevant) candidate may alter who wins the election despite the ranked preferences of all voters remaining the same. </p> <div class="mw-heading mw-heading3"><h3 id="IIA_example">IIA example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=14" title="Edit section: IIA example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a positional voting election with three candidates A, B and C where a first, second and third preference is worth 4, 2 and 1 point respectively. The 12 voters cast their ranked ballots as follows: </p> <table class="wikitable"> <tbody><tr> <th>Number of ballots</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>5 </th> <td>A</td> <td>B</td> <td>C </td></tr> <tr> <th>4 </th> <td>B</td> <td>C</td> <td>A </td></tr> <tr> <th>3 </th> <td>C</td> <td>A</td> <td>B </td></tr></tbody></table> <p>The election outcome is hence: </p> <table class="wikitable"> <tbody><tr> <th>Candidate</th> <th>Points to be tallied</th> <th>Total</th> <th>Overall rank </th></tr> <tr> <th>A </th> <td>(5 x 4) + (3 x 2) + (4 x 1)</td> <td>30</td> <td>First </td></tr> <tr> <th>B </th> <td>(4 x 4) + (5 x 2) + (3 x 1)</td> <td>29</td> <td>Second </td></tr> <tr> <th>C </th> <td>(3 x 4) + (4 x 2) + (5 x 1)</td> <td>25</td> <td>Third </td></tr></tbody></table> <p>Therefore, candidate A is the single winner and candidates B and C are the two losers. As an irrelevant alternative (loser), whether B enters the contest or not should make no difference to A winning provided the voting system is IIA compliant. </p><p>Rerunning the election without candidate B while maintaining the correct ranked preferences for A and C, the 12 ballots are now cast as follows: </p> <table class="wikitable"> <tbody><tr> <th>Number of ballots</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>5 </th> <td>A</td> <td>C</td> <td>- </td></tr> <tr> <th>4 </th> <td>C</td> <td>A</td> <td>- </td></tr> <tr> <th>3 </th> <td>C</td> <td>A</td> <td>- </td></tr></tbody></table> <p>The rerun election outcome is now: </p> <table class="wikitable"> <tbody><tr> <th>Candidate</th> <th>Points to be tallied</th> <th>Total</th> <th>Overall rank </th></tr> <tr> <th>A </th> <td>(5 x 4) + (7 x 2)</td> <td>34</td> <td>Second </td></tr> <tr> <th>C </th> <td>(7 x 4) + (5 x 2)</td> <td>38</td> <td>First </td></tr></tbody></table> <p>Given the withdrawal of candidate B, the winner is now C and no longer A. Regardless of the specific points awarded to the rank positions of the preferences, there are always some cases where the addition or deletion of an irrelevant alternative alters the outcome of an election. Hence, positional voting is not IIA compliant. </p> <div class="mw-heading mw-heading3"><h3 id="IoC_example">IoC example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=15" title="Edit section: IoC example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Positional voting also fails the <a href="/wiki/Independence_of_clones_criterion" title="Independence of clones criterion">independence of clones</a> (IoC) criterion. The <a href="/wiki/Strategic_nomination" title="Strategic nomination">strategic nomination</a> of clones is quite likely to significantly affect the outcome of an election and it is often the intention behind doing so. A clone is a nominally identical candidate to one already standing where voters are unable to distinguish between them unless informed as to which of the two is the clone. As tied rankings are not permitted, these two candidates must be ranked by voters in adjacent positions instead. Cloning may well promote or demote the collective ranking of any non-cloned candidate. </p><p>Consider a positional voting election in which three candidates may compete. There are just 12 voters and a first, second and third preference is worth 4, 2 and 1 point respectively. </p><p>In this first scenario, two candidates A and B are nominated but no clone enters the contest. The voters cast their ranked ballots as follows: </p> <table class="wikitable"> <tbody><tr> <th>Number of ballots</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>6 </th> <td>A</td> <td>B</td> <td>- </td></tr> <tr> <th>6 </th> <td>B</td> <td>A</td> <td>- </td></tr></tbody></table> <p>The election outcome is hence: </p> <table class="wikitable"> <tbody><tr> <th>Candidate</th> <th>Points to be tallied</th> <th>Total</th> <th>Overall rank </th></tr> <tr> <th>A </th> <td>(6 x 4) + (6 x 2)</td> <td>36</td> <td>First equal </td></tr> <tr> <th>B </th> <td>(6 x 4) + (6 x 2)</td> <td>36</td> <td>First equal </td></tr></tbody></table> <p>Given equal support, there is an evitable tie for first place between A and B. </p><p>Suppose B, anticipating this tie, decided to enter a clone of itself. The nominated candidates are now A, B<sub>1</sub> and B<sub>2</sub>. As the voters are unable to distinguish between B<sub>1</sub> and B<sub>2</sub>, they are just a likely to rank B<sub>1</sub> over B<sub>2</sub> as to prefer B<sub>2</sub> over B<sub>1</sub>. In this second scenario, the 12 ballots are now cast as follows: </p> <table class="wikitable"> <tbody><tr> <th>Number of ballots</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>3 </th> <td>A</td> <td>B<sub>1</sub></td> <td>B<sub>2</sub> </td></tr> <tr> <th>3 </th> <td>A</td> <td>B<sub>2</sub></td> <td>B<sub>1</sub> </td></tr> <tr> <th>3 </th> <td>B<sub>1</sub></td> <td>B<sub>2</sub></td> <td>A </td></tr> <tr> <th>3 </th> <td>B<sub>2</sub></td> <td>B<sub>1</sub></td> <td>A </td></tr></tbody></table> <p>The new election outcome is now: </p> <table class="wikitable"> <tbody><tr> <th>Candidate</th> <th>Points to be tallied</th> <th>Total</th> <th>Overall rank </th></tr> <tr> <th>A </th> <td>(6 x 4) + (0 x 2) + (6 x 1)</td> <td>30</td> <td>First </td></tr> <tr> <th>B<sub>1</sub> </th> <td>(3 x 4) + (6 x 2) + (3 x 1)</td> <td>27</td> <td>Second equal </td></tr> <tr> <th>B<sub>2</sub> </th> <td>(3 x 4) + (6 x 2) + (3 x 1)</td> <td>27</td> <td>Second equal </td></tr></tbody></table> <p>By adding a clone of itself, B has handed victory to candidate A. This counter-productive ‘spoiler’ effect or act of self-harm is called <a href="/wiki/Comparison_of_electoral_systems#Result_criteria_(relative)" class="mw-redirect" title="Comparison of electoral systems">vote-splitting</a>. </p><p>To promote itself into first place, B should instead instruct all its supporters to always prefer one of its candidates (say B<sub>1</sub>) over the other (B<sub>2</sub>). In this third scenario, the 12 ballots are now cast as follows: </p> <table class="wikitable"> <tbody><tr> <th>Number of ballots</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>3 </th> <td>A</td> <td>B<sub>1</sub></td> <td>B<sub>2</sub> </td></tr> <tr> <th>3 </th> <td>A</td> <td>B<sub>2</sub></td> <td>B<sub>1</sub> </td></tr> <tr> <th>6 </th> <td>B<sub>1</sub></td> <td>B<sub>2</sub></td> <td>A </td></tr></tbody></table> <p>The revised election outcome is now: </p> <table class="wikitable"> <tbody><tr> <th>Candidate</th> <th>Points to be tallied</th> <th>Total</th> <th>Overall rank </th></tr> <tr> <th>A </th> <td>(6 x 4) + (0 x 2) + (6 x 1)</td> <td>30</td> <td>Second </td></tr> <tr> <th>B<sub>1</sub> </th> <td>(6 x 4) + (3 x 2) + (3 x 1)</td> <td>33</td> <td>First </td></tr> <tr> <th>B<sub>2</sub> </th> <td>(0 x 4) + (9 x 2) + (3 x 1)</td> <td>21</td> <td>Third </td></tr></tbody></table> <p>By ‘team’ B signalling to its own supporters - but not to A supporters - which of its two candidates it wants to win, B has achieved its objective of gaining victory for B<sub>1</sub>. With no clone, A and B tie with equal numbers of first and second preferences. The introduction of clone B<sub>2</sub> (an irrelevant alternative) has pushed the second preferences for A into third place while preferences for ‘team’ B (B or B<sub>1</sub>) are unchanged in the first and third scenarios. This wilful act to ‘bury’ A and promote itself is called <a href="/wiki/Comparison_of_electoral_systems#Result_criteria_(relative)" class="mw-redirect" title="Comparison of electoral systems">teaming</a>. Note that if A signals to its own supporters to always prefer B<sub>2</sub> over B<sub>1</sub> in a tit-for-tat retaliation then the original tie between A and ‘team’ B is re-established. </p><p>To a greater or lesser extent, all positional voting systems are vulnerable to teaming; with the sole exception of a plurality-equivalent one. As only first preferences have any value, employing clones to ‘bury’ opponents down in rank never affects election outcomes. However, precisely because only first preferences have any value, plurality is instead particularly susceptible to vote-splitting. To a lesser extent, many other positional voting systems are also affected by ‘spoiler’ candidates. While inherently vulnerable to teaming, the Borda count is however invulnerable to vote-splitting.<sup id="cite_ref-Saari_1-5" class="reference"><a href="#cite_note-Saari-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Donald_G._Saari" title="Donald G. Saari">Donald G. Saari</a> has published various works that mathematically analyse positional voting electoral systems. The fundamental method explored in his analysis is the Borda count. </p> <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=Positional_voting&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Saari-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Saari_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Saari_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Saari_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Saari_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Saari_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Saari_1-5"><sup><i><b>f</b></i></sup></a></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="CITEREFSaari1995" class="citation book cs1">Saari, Donald G. (1995). <i>Basic Geometry of Voting</i>. Springer-Verlag. pp. 101–103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-60064-7" title="Special:BookSources/3-540-60064-7"><bdi>3-540-60064-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Geometry+of+Voting&rft.pages=101-103&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=3-540-60064-7&rft.aulast=Saari&rft.aufirst=Donald+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APositional+voting" class="Z3988"></span></span> </li> <li id="cite_note-Nauru_2019-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nauru_2019_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://election.com.nr/wp-content/uploads/2019/12/2019-Parliamentary-Election-Report.pdf">"2019 Parliamentary Election Final Report"</a> <span class="cs1-format">(PDF)</span>. NAOERO Electoral Commission<span class="reference-accessdate">. Retrieved <span class="nowrap">4 November</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=2019+Parliamentary+Election+Final+Report&rft.pub=NAOERO+Electoral+Commission&rft_id=https%3A%2F%2Felection.com.nr%2Fwp-content%2Fuploads%2F2019%2F12%2F2019-Parliamentary-Election-Report.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APositional+voting" class="Z3988"></span></span> </li> <li id="cite_note-Nauru_2022-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nauru_2022_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://election.com.nr/wp-content/uploads/2023/03/2022-PE-Final-Report-web2.pdf">"2022 Parliamentary Election Final Report"</a> <span class="cs1-format">(PDF)</span>. NAOERO Electoral Commission<span class="reference-accessdate">. Retrieved <span class="nowrap">4 November</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=2022+Parliamentary+Election+Final+Report&rft.pub=NAOERO+Electoral+Commission&rft_id=https%3A%2F%2Felection.com.nr%2Fwp-content%2Fuploads%2F2023%2F03%2F2022-PE-Final-Report-web2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APositional+voting" class="Z3988"></span></span> </li> <li id="cite_note-Reilly20022-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Reilly20022_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReilly2002" class="citation journal cs1">Reilly, Benjamin (2002). "Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries". <i>International Political Science Review</i>. <b>23</b> (4): 364–366. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.924.3992">10.1.1.924.3992</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F0192512102023004002">10.1177/0192512102023004002</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3213336">3213336</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Political+Science+Review&rft.atitle=Social+Choice+in+the+South+Seas%3A+Electoral+Innovation+and+the+Borda+Count+in+the+Pacific+Island+Countries&rft.volume=23&rft.issue=4&rft.pages=364-366&rft.date=2002&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.924.3992%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3213336%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1177%2F0192512102023004002&rft.aulast=Reilly&rft.aufirst=Benjamin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APositional+voting" class="Z3988"></span></span> </li> <li id="cite_note-Fraenkel_Grofman_20142-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fraenkel_Grofman_20142_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fraenkel_Grofman_20142_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Fraenkel_Grofman_20142_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraenkelGrofman2014" class="citation journal cs1">Fraenkel, Jon; Grofman, Bernard (2014-04-03). "The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia". <i>Australian Journal of Political Science</i>. <b>49</b> (2): 186–205. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10361146.2014.900530">10.1080/10361146.2014.900530</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:153325225">153325225</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Australian+Journal+of+Political+Science&rft.atitle=The+Borda+Count+and+its+real-world+alternatives%3A+Comparing+scoring+rules+in+Nauru+and+Slovenia&rft.volume=49&rft.issue=2&rft.pages=186-205&rft.date=2014-04-03&rft_id=info%3Adoi%2F10.1080%2F10361146.2014.900530&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A153325225%23id-name%3DS2CID&rft.aulast=Fraenkel&rft.aufirst=Jon&rft.au=Grofman%2C+Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3APositional+voting" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Positional_voting&action=edit&section=18" 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://ssrn.com/abstract=195769">Economic Theory, Vol. 15, Issue 1, 2000: <i>Mathematical Structure of Voting Paradoxes: II. Positional Voting</i>, Donald G. SAARI</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #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 .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a> <ul><li><a href="/wiki/Combined_approval_voting" title="Combined approval voting">Combined approval voting</a></li> <li><a href="/wiki/Unified_primary" title="Unified primary">Unified primary</a></li></ul></li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a></li> <li><a href="/wiki/Bucklin_voting" title="Bucklin voting">Bucklin voting</a></li> <li><a href="/wiki/Condorcet_methods" class="mw-redirect" title="Condorcet methods">Condorcet methods</a> <ul><li><a href="/wiki/Copeland%27s_method" title="Copeland's method">Copeland's method</a></li> <li><a href="/wiki/Dodgson%27s_method" title="Dodgson's method">Dodgson's method</a></li> <li><a href="/wiki/Kemeny%E2%80%93Young_method" title="Kemeny–Young method">Kemeny–Young method</a></li> <li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax Condorcet method</a></li> <li><a href="/wiki/Nanson%27s_method" title="Nanson's method">Nanson's method</a></li> <li><a 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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