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class="mw-page-title-main">Proportional approval voting</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vote_d%27approbation_proportionnel" title="Vote d'approbation proportionnel – French" lang="fr" hreflang="fr" data-title="Vote d'approbation proportionnel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Proporcjonalna_metoda_g%C5%82osowania_przez_aprobaty" title="Proporcjonalna metoda głosowania przez aprobaty – Polish" lang="pl" hreflang="pl" data-title="Proporcjonalna metoda głosowania przez aprobaty" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A5%E0%B8%87%E0%B8%84%E0%B8%B0%E0%B9%81%E0%B8%99%E0%B8%99%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%84%E0%B8%B0%E0%B9%81%E0%B8%99%E0%B8%99%E0%B8%AD%E0%B8%99%E0%B8%B8%E0%B8%A1%E0%B8%B1%E0%B8%95%E0%B8%B4%E0%B9%80%E0%B8%9B%E0%B9%87%E0%B8%99%E0%B8%AA%E0%B8%B1%E0%B8%94%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99" title="การลงคะแนนแบบคะแนนอนุมัติเป็นสัดส่วน – Thai" lang="th" hreflang="th" data-title="การลงคะแนนแบบคะแนนอนุมัติเป็นสัดส่วน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AF%94%E4%BE%8B%E6%89%B9%E5%87%86%E6%8A%95%E7%A5%A8%E5%88%B6" title="比例批准投票制 – Chinese" lang="zh" hreflang="zh" data-title="比例批准投票制" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Multiple-winner electoral system</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">For an overview of proportional variants of approval voting, see <a href="/wiki/Multiwinner_approval_voting" title="Multiwinner approval voting">multiwinner approval voting</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">A joint <a href="/wiki/Portal:Politics" title="Portal:Politics">Politics</a> and <a href="/wiki/Portal:Economics" title="Portal:Economics">Economics</a> series</td></tr><tr><th class="sidebar-title-with-pretitle" style="border-top:1px #fafafa solid; border-bottom:1px #fafafa solid; background:#efefef; background: var(--background-color-interactive, #efefef); color: var(--color-base, #000); padding:0.2em;"><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice</a> and <a href="/wiki/Electoral_system" title="Electoral system">electoral systems</a></th></tr><tr><td class="sidebar-image"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Electoral-systems-gears.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/128px-Electoral-systems-gears.svg.png" decoding="async" width="128" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/192px-Electoral-systems-gears.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/256px-Electoral-systems-gears.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption></figcaption></figure></td></tr><tr><td class="sidebar-above"> <div class="hlist"><ul><li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice</a></li><li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li><li><a href="/wiki/Comparative_politics" title="Comparative politics">Comparative politics</a></li><li><a href="/wiki/Comparison_of_voting_rules" title="Comparison of voting rules">Comparison</a></li><li><a href="/wiki/List_of_electoral_systems" title="List of electoral systems">List</a> (<a href="/wiki/List_of_electoral_systems_by_country" title="List of electoral systems by country">By country</a>)</li></ul></div></td></tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Single-member_district" title="Single-member district">Single-winner methods</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Single vote - <a href="/wiki/Plurality_voting" title="Plurality voting">plurality</a> methods</b> <ul><li><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First preference plurality (FPP)</a></li> <li><a href="/wiki/Two-round_system" title="Two-round system">Two-round</a> (<abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> <a href="/wiki/Nonpartisan_blanket_primary" title="Nonpartisan blanket primary">Jungle primary</a>) <ul><li><a href="/wiki/Partisan_primary" class="mw-redirect" title="Partisan primary">Partisan primary</a></li></ul></li> <li><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> Alternative vote (AV)</li> <li><abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> Ranked-choice (RCV)</li></ul></li></ul> <hr /> <p><b><a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet methods</a></b> </p> <ul><li><a href="/wiki/Tideman_alternative_method" title="Tideman alternative method">Condorcet-IRV</a></li> <li><a href="/wiki/Round-robin_voting" title="Round-robin voting">Round-robin voting</a> <ul><li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax</a></li> <li><a href="/wiki/Schulze_method" title="Schulze method">Schulze</a></li> <li><a href="/wiki/Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Positional_voting" title="Positional voting">Positional voting</a></b> </p> <ul><li><a href="/wiki/First-preference_plurality" class="mw-redirect" title="First-preference plurality">Plurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">IRV</a>)</li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Baldwin%27s_method" class="mw-redirect" title="Baldwin's method">Baldwin</a>)</li> <li><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Antiplurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Coombs_method" class="mw-redirect" title="Coombs method">Coombs</a>)</li></ul> <hr /> <p><b><a href="/wiki/Rated_voting" title="Rated voting">Cardinal voting</a></b> </p> <ul><li><a href="/wiki/Score_voting" title="Score voting">Score voting</a></li> <li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a></li> <li><a href="/wiki/Highest_median_voting_rules" title="Highest median voting rules">Majority judgment</a></li> <li><a href="/wiki/STAR_voting" title="STAR voting">STAR voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Proportional_representation" title="Proportional representation">Proportional representation</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></b> <ul><li><a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a> <ul><li><a href="/wiki/Highest_averages_method" title="Highest averages method">Highest averages</a></li> <li><a href="/wiki/Largest_remainder_method" class="mw-redirect" title="Largest remainder method">Largest remainders</a></li> <li><a href="/wiki/National_remnant" title="National remnant">National remnant</a></li> <li><a href="/wiki/Biproportional_apportionment" title="Biproportional apportionment">Biproportional</a></li></ul></li> <li><a href="/wiki/Electoral_list" title="Electoral list">List type</a> <ul><li><a href="/wiki/Closed_list" title="Closed list">Closed list</a></li> <li><a href="/wiki/Open_list" title="Open list">Open list</a></li> <li><a href="/wiki/Panachage" title="Panachage">Panachage</a></li> <li><a href="/wiki/Justified_representation" title="Justified representation">List-free PR</a></li> <li><a href="/wiki/Localized_list" title="Localized list">Localized list</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b> </p> <ul><li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Hare STV</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li> <li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul> <hr /> <p><b><a href="/wiki/Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b> </p> <ul><li><a class="mw-selflink selflink">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 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template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Proportional approval voting</b> (<b>PAV</b>) is a <a href="/wiki/Proportional_representation" title="Proportional representation">proportional</a> <a href="/wiki/Electoral_system" title="Electoral system">electoral system</a> for <a href="/wiki/Multiwinner_elections" class="mw-redirect" title="Multiwinner elections">multiwinner elections</a>. It is a <a href="/wiki/Multiwinner_approval_voting" title="Multiwinner approval voting">multiwinner approval method</a> that extends the <a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt method</a> of <a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">apportionment</a> commonly used to calculate apportionments for <a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">party-list proportional representation</a>.<sup id="cite_ref-dhondt_1-0" class="reference"><a href="#cite_note-dhondt-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, PAV allows voters to support only the candidates they approve of, rather than being forced to approve or reject all candidates on a given <a href="/wiki/Electoral_list" title="Electoral list">party list</a>.<sup id="cite_ref-pavaxioms_2-0" class="reference"><a href="#cite_note-pavaxioms-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In PAV, voters cast <a href="/wiki/Approval_ballot" title="Approval ballot">approval ballots</a> marking all candidates they approve of; each voter's ballot is then treated as if all candidates on the ballot were on their own "party list." Seats are then <a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">apportioned</a> between candidates in a way that ensures all coalitions are represented proportionally. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>PAV is a special case of <a href="/wiki/Thiele%27s_voting_rules" title="Thiele's voting rules">Thiele's voting rule</a>, proposed by <a href="/wiki/Thorvald_N._Thiele" title="Thorvald N. Thiele">Thorvald N. Thiele</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jansonsurvey_4-0" class="reference"><a href="#cite_note-jansonsurvey-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It was used in combination with <a href="/wiki/Ranked_voting" title="Ranked voting">ranked voting</a> in the Swedish elections from 1909 to 1921 for distributing seats within parties and in local elections.<sup id="cite_ref-jansonsurvey_4-1" class="reference"><a href="#cite_note-jansonsurvey-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> PAV was rediscovered by Forest Simmons in 2001,<sup id="cite_ref-Kilgour2010_5-0" class="reference"><a href="#cite_note-Kilgour2010-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> who gave it the name "proportional approval voting." </p> <div class="mw-heading mw-heading2"><h2 id="Method">Method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=2" title="Edit section: Method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Highest_averages_method" title="Highest averages method">Highest averages method</a></div> <p>Like its close cousin, <a href="/wiki/Satisfaction_approval_voting" title="Satisfaction approval voting">satisfaction approval voting</a>, PAV can be thought of as selecting a committee by testing all possible committees, then choosing the committee with the most votes. In satisfaction approval voting, each voter's ballot is split equally between all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> candidates they approve of, giving each one <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab96580d23ec5eff6bb0e666531eccb7a8035d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.374ex; height:2.843ex;" alt="{\displaystyle 1/r}"></span> votes. If voters are perfectly strategic, and only support as many candidates as their party is entitled to, SAV creates a proportional result. </p><p>PAV makes one modification to remove this need for strategy: it only splits a voter's ballot <i>after</i> they have elected a candidate. As a result, voters can freely approve of losing candidates without diluting their ballot. Voters contribute a whole vote to the first candidate they support who is elected; half a vote to the second candidate; and so on. </p><p>Thus, of a ballot approves of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> candidates who are elected, that ballot contributes the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>-th <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a> to that committee's vote total. In other words:<sup id="cite_ref-Kilgour2010_5-1" class="reference"><a href="#cite_note-Kilgour2010-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-justified-representation_6-0" class="reference"><a href="#cite_note-justified-representation-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {H} (r)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} (r)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8b1604a997dc1fad9444a6f92062357fedcebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.943ex; height:5.176ex;" alt="{\displaystyle \mathrm {H} (r)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{r}}}"></span></dd></dl> <p>The score for a given committee is calculated as the sum of the scores garnered from all the voters. We then choose the committee with the highest score. </p><p>Formally, assume we have a set of candidates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\{c_{1},c_{2},\ldots ,c_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\{c_{1},c_{2},\ldots ,c_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e798e4008e1779c26ccb19f8bf6ddd8acdd0ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.206ex; height:2.843ex;" alt="{\displaystyle C=\{c_{1},c_{2},\ldots ,c_{m}\}}"></span>, a set of voters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\{1,2,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46de4999ceee34d1bf6626160ea4528696de1af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.419ex; height:2.843ex;" alt="{\displaystyle N=\{1,2,\ldots ,n\}}"></span>, and a committee size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> denote the set of candidates approved by voter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98a794279966819df68e268cf26198f9f5b32f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.707ex; height:2.176ex;" alt="{\displaystyle i\in N}"></span>. The PAV score of a committee <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subseteq C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⊆</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\subseteq C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e060f5f38a040c5654b78d0cdadca15c5159c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.3ex; height:2.343ex;" alt="{\displaystyle W\subseteq C}"></span> with size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |W|=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |W|=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a01d6ec18a5b9f531e77a927876c44f59c4f912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.039ex; height:2.843ex;" alt="{\displaystyle |W|=k}"></span> is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathrm {sc} _{\mathrm {PAV} }(W)=\sum _{i=1}^{n}\mathrm {H} (|A_{i}\cap W|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathrm {sc} _{\mathrm {PAV} }(W)=\sum _{i=1}^{n}\mathrm {H} (|A_{i}\cap W|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74b997c40dfcbf28f1fdef26e3763e34fcd2e65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.256ex; height:3.176ex;" alt="{\displaystyle \textstyle \mathrm {sc} _{\mathrm {PAV} }(W)=\sum _{i=1}^{n}\mathrm {H} (|A_{i}\cap W|)}"></span>. PAV selects the committee <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> with the maximal score. </p> <div class="mw-heading mw-heading3"><h3 id="Example_1">Example 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=3" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assume 2 seats to be filled, and there are four candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D), and 30 voters. The ballots are: </p> <ul><li>5 voters voted for A and B</li> <li>17 voters voted for A and C</li> <li>8 voters voted for D</li></ul> <p>There are 6 possible results: AB, AC, AD, BC, BD, and CD. </p> <table class="wikitable"> <tbody><tr> <th></th> <th>AB</th> <th>AC</th> <th>AD</th> <th>BC</th> <th>BD</th> <th>CD </th></tr> <tr> <td>score from the 5 voters voting for AB</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot \left(1+{\tfrac {1}{2}}\right)=7.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>7.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot \left(1+{\tfrac {1}{2}}\right)=7.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2940e6eab675784c813506ae6abb54329a48079d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.703ex; height:3.509ex;" alt="{\displaystyle 5\cdot \left(1+{\tfrac {1}{2}}\right)=7.5}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 1=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 1=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78992e9318f873d145ca4beae0a5979837230615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 5\cdot 1=5}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 1=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 1=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78992e9318f873d145ca4beae0a5979837230615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 5\cdot 1=5}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 1=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 1=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78992e9318f873d145ca4beae0a5979837230615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 5\cdot 1=5}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 1=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 1=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78992e9318f873d145ca4beae0a5979837230615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 5\cdot 1=5}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b919679d07d0186ab4d5c42eb418ddf76ec8bd3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 5\cdot 0=0}"></span> </td></tr> <tr> <td>score from the 17 voters voting for AC</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17\cdot 1=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17\cdot 1=17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba55995a3267cacd1124b340f1bb9919981aa09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.59ex; height:2.176ex;" alt="{\displaystyle 17\cdot 1=17}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17\cdot \left(1+{\tfrac {1}{2}}\right)=25.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>25.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17\cdot \left(1+{\tfrac {1}{2}}\right)=25.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27ba4a0e1e23710a813918721aebb4dd44f7c3f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.027ex; height:3.509ex;" alt="{\displaystyle 17\cdot \left(1+{\tfrac {1}{2}}\right)=25.5}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17\cdot 1=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17\cdot 1=17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba55995a3267cacd1124b340f1bb9919981aa09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.59ex; height:2.176ex;" alt="{\displaystyle 17\cdot 1=17}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17\cdot 1=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17\cdot 1=17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba55995a3267cacd1124b340f1bb9919981aa09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.59ex; height:2.176ex;" alt="{\displaystyle 17\cdot 1=17}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c4512429efafcd1bbbdec86ab4a00793b76b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.427ex; height:2.176ex;" alt="{\displaystyle 17\cdot 0=0}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17\cdot 1=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17\cdot 1=17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba55995a3267cacd1124b340f1bb9919981aa09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.59ex; height:2.176ex;" alt="{\displaystyle 17\cdot 1=17}"></span> </td></tr> <tr> <td>score from the 8 voters voting for D</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d88e94a67216d9dd8b99726b74a03637c163fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 8\cdot 0=0}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d88e94a67216d9dd8b99726b74a03637c163fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 8\cdot 0=0}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 1=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\cdot 1=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4abdd761134aecbedc7890b0edb1ad931641387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 8\cdot 1=8}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d88e94a67216d9dd8b99726b74a03637c163fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 8\cdot 0=0}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 1=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\cdot 1=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4abdd761134aecbedc7890b0edb1ad931641387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 8\cdot 1=8}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 1=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\cdot 1=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4abdd761134aecbedc7890b0edb1ad931641387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 8\cdot 1=8}"></span> </td></tr> <tr> <td>Total score</td> <td>24.5</td> <td><b>30.5</b></td> <td>30</td> <td>22</td> <td>13</td> <td>25 </td></tr></tbody></table> <p>Andrea and Carter are elected. </p><p>Note that Simple Approval shows that Andrea has 22 votes, Carter has 17 votes, Delilah has 8 votes and Brad has 5 votes. In this case, the PAV selection of Andrea and Carter is coincident with the Simple Approval sequence. However, if the above votes are changed slightly so that A and C receive 16 votes and D receives 9 votes, then Andrea and Delilah are elected since the A and C score is now 29 while the A and D score remains at 30. Also, the sequence created by using Simple Approval remains unchanged. This shows that PAV can give results that are incompatible with the method which simply follows the sequence implied by Simple Approval. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=4" title="Edit section: Example 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assume there are 10 seats to be selected, and there are three groups of candidates: <span style="color:red">red</span>, <span style="color:blue">blue</span>, and <span style="color:#228B22">green</span> candidates. There are 100 voters: </p> <ul><li><span style="color:blue">60 voters</span> voted for all <span style="color:blue">blue</span> candidates,</li> <li><span style="color:red">30 voters</span> voted for all <span style="color:red">red</span> candidates,</li> <li><span style="color:#228B22">10 voters</span> voted for all <span style="color:#228B22">green</span> candidates.</li></ul> <p>In this case PAV would select <span style="color:blue">6 blue</span>, <span style="color:red">3 red</span>, and <span style="color:#228B22">1 green</span> candidate. The score of such a committee would be<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 60\cdot \mathrm {H} (6)+30\cdot \mathrm {H} (3)+10\cdot \mathrm {H} (1)=60\cdot \left(1+{\tfrac {1}{2}}+\ldots +{\tfrac {1}{6}}\right)+30\cdot \left(1+{\tfrac {1}{2}}+{\tfrac {1}{3}}\right)+10\cdot 1=147+55+10=212{\text{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>60</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>30</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>10</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>60</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>30</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>10</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>147</mn> <mo>+</mo> <mn>55</mn> <mo>+</mo> <mn>10</mn> <mo>=</mo> <mn>212</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60\cdot \mathrm {H} (6)+30\cdot \mathrm {H} (3)+10\cdot \mathrm {H} (1)=60\cdot \left(1+{\tfrac {1}{2}}+\ldots +{\tfrac {1}{6}}\right)+30\cdot \left(1+{\tfrac {1}{2}}+{\tfrac {1}{3}}\right)+10\cdot 1=147+55+10=212{\text{.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6076f1f3511bbbc7b61387edbf037b090bc781f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:109.374ex; height:4.843ex;" alt="{\displaystyle 60\cdot \mathrm {H} (6)+30\cdot \mathrm {H} (3)+10\cdot \mathrm {H} (1)=60\cdot \left(1+{\tfrac {1}{2}}+\ldots +{\tfrac {1}{6}}\right)+30\cdot \left(1+{\tfrac {1}{2}}+{\tfrac {1}{3}}\right)+10\cdot 1=147+55+10=212{\text{.}}}"></span>All other committees receive a lower score. For example, the score of a committee that consists of only blue candidates would be </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60\cdot \mathrm {H} (10)+30\cdot \mathrm {H} (0)+10\cdot \mathrm {H} (0)=60\cdot \left(1+{\tfrac {1}{2}}+\ldots +{\tfrac {1}{10}}\right)\approx 175.73{\text{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>60</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>30</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>10</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>60</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>175.73</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60\cdot \mathrm {H} (10)+30\cdot \mathrm {H} (0)+10\cdot \mathrm {H} (0)=60\cdot \left(1+{\tfrac {1}{2}}+\ldots +{\tfrac {1}{10}}\right)\approx 175.73{\text{.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4bdb8d6823deeda696b6fbf105a0e79000342d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:69.627ex; height:4.843ex;" alt="{\displaystyle 60\cdot \mathrm {H} (10)+30\cdot \mathrm {H} (0)+10\cdot \mathrm {H} (0)=60\cdot \left(1+{\tfrac {1}{2}}+\ldots +{\tfrac {1}{10}}\right)\approx 175.73{\text{.}}}"></span></dd></dl> <p>In this example, the outcome of PAV is <a href="/wiki/Proportional_representation" title="Proportional representation">proportional</a>: the number of candidates selected from each group is proportional to the number of voters voting for the group. This is not coincidence: If the candidates form disjoint groups, as in the above example (the groups can be viewed as political parties), and each voter votes exclusively for all of the candidates within a single group, then PAV will act in the same way as the <a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt method</a> of <a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">party-list proportional representation</a>.<sup id="cite_ref-dhondt_1-1" class="reference"><a href="#cite_note-dhondt-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section describes axiomatic properties of Proportional Approval Voting. </p> <div class="mw-heading mw-heading3"><h3 id="Committees_of_size_one">Committees of size one</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=6" title="Edit section: Committees of size one"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In an election with only one winner, PAV operates in exactly the same way as <a href="/wiki/Approval_voting" title="Approval voting">approval voting</a>. That is, it selects the committee consisting of the candidate who is approved by the most voters. </p> <div class="mw-heading mw-heading3"><h3 id="Proportionality">Proportionality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=7" title="Edit section: Proportionality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most systems of <a href="/wiki/Proportional_representation" title="Proportional representation">proportional representation</a> use <a href="/wiki/Party-list_system" title="Party-list system">party lists</a>. PAV was designed to have both proportional representation and personal votes (voters vote for candidates, not for a party list). It deserves to be called a "proportional" system because if votes turn out to follow a partisan scheme (each voter votes for all candidates from a party and no other) then the system elects a number of candidates in each party that is proportional to the number of voters who chose this party (see <a href="#Example_2">Example 2</a>).<sup id="cite_ref-dhondt_1-2" class="reference"><a href="#cite_note-dhondt-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Further, under mild assumptions (symmetry, continuity and <a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a>), PAV is the only extension of the <a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt method</a> that allows personal votes and satisfies the <a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">consistency criterion</a>.<sup id="cite_ref-pavaxioms_2-1" class="reference"><a href="#cite_note-pavaxioms-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Even if the voters do not follow the partisan scheme, the rule provides proportionality guarantees. For example, PAV satisfies the strong fairness property called <a href="/wiki/Extended_Justified_representation" class="mw-redirect" title="Extended Justified representation">extended justified representation</a>,<sup id="cite_ref-justified-representation_6-1" class="reference"><a href="#cite_note-justified-representation-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> as well as the related property <a href="/wiki/Proportional_justified_representation" class="mw-redirect" title="Proportional justified representation">proportional justified representation</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> It also has optimal proportionality degree.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> All these properties guarantee that any group of voters with cohesive (similar) preferences will be represented by a number of candidates that is at least proportional to the size of the group. PAV is the only method satisfying such properties among all PAV-like optimization methods (that may use numbers other than harmonic numbers in their definition).<sup id="cite_ref-justified-representation_6-2" class="reference"><a href="#cite_note-justified-representation-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The committees returned by PAV might not be in the <a href="/wiki/Core_(game_theory)" title="Core (game theory)">core</a>.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style#Technical_language" title="Wikipedia:Manual of Style"><span title="The material near this tag may be using jargon that limits the article's accessibility. (May 2024)">jargon</span></a></i>]</sup><sup id="cite_ref-justified-representation_6-3" class="reference"><a href="#cite_note-justified-representation-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-welfarismLimits_10-0" class="reference"><a href="#cite_note-welfarismLimits-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> However, it guarantees 2-approximation of the core,<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style#Technical_language" title="Wikipedia:Manual of Style"><span title="The material near this tag may be using jargon that limits the article's accessibility. (May 2024)">jargon</span></a></i>]</sup> which is the optimal approximation ratio that can be achieved by a rule satisfying the <a href="/wiki/Pigou%E2%80%93Dalton_principle" title="Pigou–Dalton principle">Pigou–Dalton principle</a> of transfers.<sup id="cite_ref-welfarismLimits_10-1" class="reference"><a href="#cite_note-welfarismLimits-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Furthermore, PAV satisfies the property of the core if there are sufficiently many similar candidates running in an election.<sup id="cite_ref-approval-apportionment_11-0" class="reference"><a href="#cite_note-approval-apportionment-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>PAV fails priceability (that is, the choice of PAV cannot be always explained via a process where the voters are endowed with a fixed amount of virtual money, and spend this spend money on buying candidates they like) and fails laminar proportionality.<sup id="cite_ref-welfarismLimits_10-2" class="reference"><a href="#cite_note-welfarismLimits-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Two alternative rules that satisfy priceability and laminar proportionality, and that have comparably good proportionality-related properties to PAV are the <a href="/wiki/Method_of_equal_shares" title="Method of equal shares">method of equal shares</a> and <a href="/wiki/Phragmen%27s_voting_rules" title="Phragmen's voting rules">Phragmén's sequential rules</a>.<sup id="cite_ref-abcsurvey_12-0" class="reference"><a href="#cite_note-abcsurvey-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> These two alternative methods are also computable in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>, yet they fail <a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_properties">Other properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=8" title="Edit section: Other properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Apart from properties pertaining to proportionality, PAV <b>satisfies</b> the following axioms: </p> <ul><li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li> <li><a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">Consistency</a></li> <li>Support monotonicity (if the support of a winning candidate increases, i.e., more voters vote for this candidate, then this candidate remains winning)<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Pigou%E2%80%93Dalton_principle" title="Pigou–Dalton principle">Pigou–Dalton principle</a><sup id="cite_ref-welfarismLimits_10-3" class="reference"><a href="#cite_note-welfarismLimits-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul> <p>PAV <b>fails</b> the following properties: </p> <ul><li><a href="/wiki/House_monotonicity" title="House monotonicity">House monotonicity</a>.<sup id="cite_ref-abcsurvey_12-1" class="reference"><a href="#cite_note-abcsurvey-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Two alternative methods that satisfy house monotonicity and that have comparably good proportionality-related properties to PAV are <a href="/wiki/Sequential_proportional_approval_voting" title="Sequential proportional approval voting">Sequential Proportional Approval Voting</a> and <a href="/wiki/Phragmen%27s_voting_rules" title="Phragmen's voting rules">Phragmén's Sequential Rules</a>.<sup id="cite_ref-abcsurvey_12-2" class="reference"><a href="#cite_note-abcsurvey-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> These two alternative methods are also computable in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>, but they fail <a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a>, <a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">Consistency</a>, and the <a href="/wiki/Pigou%E2%80%93Dalton_principle" title="Pigou–Dalton principle">Pigou–Dalton principle</a>.<sup id="cite_ref-abcsurvey_12-3" class="reference"><a href="#cite_note-abcsurvey-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=9" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:ILP-PAV.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2a/ILP-PAV.jpg" decoding="async" width="303" height="165" class="mw-file-element" data-file-width="303" data-file-height="165" /></a><figcaption>An <a href="/wiki/Integer_linear_programming" class="mw-redirect" title="Integer linear programming">integer linear programming</a> formulation for computing winning committees according to PAV. The variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef486676b03c4281283a2a6aaa304c0f90fa54bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.083ex; height:2.009ex;" alt="{\displaystyle y_{c}}"></span> indicates whether candidate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is selected or not. The variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i,\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i,\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a250666c99f3f97d920163f01d46a0ce568269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.272ex; height:2.343ex;" alt="{\displaystyle x_{i,\ell }}"></span> indicates whether voter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> approves at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> selected candidates.<sup id="cite_ref-abcsurvey_12-4" class="reference"><a href="#cite_note-abcsurvey-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_14-0" class="reference"><a href="#cite_note-:0-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>PAV solutions and their quality can be verified in <a href="/wiki/Polynomial-time" class="mw-redirect" title="Polynomial-time">polynomial-time</a>,<sup id="cite_ref-AzizGaspers2014_15-0" class="reference"><a href="#cite_note-AzizGaspers2014-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pav-approx_16-0" class="reference"><a href="#cite_note-pav-approx-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> making transparency easy. However, the <a href="/wiki/Worst-case" class="mw-redirect" title="Worst-case">worst-case</a> time complexity is <a href="/wiki/NP-completeness" title="NP-completeness">NP-complete</a>, meaning that for some elections it can be difficult or impossible to find an exact solution that guarantees all the theoretical properties of PAV. </p><p>In practice, the outcome of PAV can be computed exactly for medium-sized committees (<50 candidates) using <a href="/wiki/Integer_programming" title="Integer programming">integer programming</a> solvers (such as those provided in the <a rel="nofollow" class="external text" href="https://github.com/martinlackner/abcvoting"><i>abcvoting</i></a> Python package). Finding an exact solution has time complexity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(nm^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(nm^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1c674dc7e56bfacd868412d2cd20fab50f6838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.106ex; height:3.176ex;" alt="{\displaystyle O(nm^{k})}"></span> with <span class="texhtml"><i>k</i></span> seats and <span class="texhtml"><i>n</i></span> voters. </p><p>From the perspective of <a href="/wiki/Parameterized_complexity" title="Parameterized complexity">parameterized complexity</a>, the problem of computing PAV is theoretically difficult outside of a few exceptional easy cases.<sup id="cite_ref-AzizGaspers2014_15-1" class="reference"><a href="#cite_note-AzizGaspers2014-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Luckily, such cases are often good approximations of real elections, allowing them to be used as heuristics that dramatically reduce the computational effort of finding a correct solution. For example, exact election results can be solved in polynomial time in the case where voters and candidates lie along a single-dimensional political spectrum,<sup id="cite_ref-:0_14-1" class="reference"><a href="#cite_note-:0-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and in linear time when voters are strong partisans (i.e. vote for party lists instead of candidates). </p> <div class="mw-heading mw-heading3"><h3 id="Deterministic_approximations">Deterministic approximations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=10" title="Edit section: Deterministic approximations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Sequential_proportional_approval_voting" title="Sequential proportional approval voting">Sequential proportional approval voting</a> is a greedy approximation for PAV with a worst-case approximation ratio of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-1/e\approx 0.63}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> <mo>≈</mo> <mn>0.63</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-1/e\approx 0.63}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec492ff4098579a000574aeed0f69f3f00c71163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.644ex; height:2.843ex;" alt="{\displaystyle 1-1/e\approx 0.63}"></span>, so the PAV score of the resulting committee is at least 63% of the optimal.<sup id="cite_ref-pav-approx_16-1" class="reference"><a href="#cite_note-pav-approx-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> This method can be computed in polynomial time, and the election outcome could potentially be determined by hand. A different approach including <a href="/wiki/Randomized_rounding" title="Randomized rounding">derandomized rounding</a> (with the <a href="/wiki/Method_of_conditional_probabilities" title="Method of conditional probabilities">method of conditional probabilities</a>) gives a worst-case approximation ratio of 0.7965;<sup id="cite_ref-pav-approx2_19-0" class="reference"><a href="#cite_note-pav-approx2-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> under standard assumptions in complexity theory, this is the best approximation ratio that can be achieved for PAV in polynomial time.<sup id="cite_ref-pav-approx2_19-1" class="reference"><a href="#cite_note-pav-approx2-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The problem of approximating PAV can be also formulated as a minimization problem (instead of maximizing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathrm {sc} _{\mathrm {PAV} }(W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathrm {sc} _{\mathrm {PAV} }(W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d420017d90326c145a8895ff9f741589f158b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.01ex; height:2.843ex;" alt="{\displaystyle \textstyle \mathrm {sc} _{\mathrm {PAV} }(W)}"></span> we want to minimize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i\in N}\mathrm {H} (k)-\mathrm {sc} _{\mathrm {PAV} }(W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i\in N}\mathrm {H} (k)-\mathrm {sc} _{\mathrm {PAV} }(W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f0c0025e3fad83a97fb56d0d0e34198241ecb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.81ex; height:3.009ex;" alt="{\displaystyle \textstyle \sum _{i\in N}\mathrm {H} (k)-\mathrm {sc} _{\mathrm {PAV} }(W)}"></span>), in which case the best known approximation ratio is 2.36.<sup id="cite_ref-pav-approx3_20-0" class="reference"><a href="#cite_note-pav-approx3-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFLacknerSkowron2023" class="citation book cs1">Lackner, Martin; Skowron, Piotr (2023). <i>Multi-Winner Voting with Approval Preferences</i>. SpringerBriefs in Intelligent Systems. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.01795">2007.01795</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/10.1007%2F978-3-031-09016-5">10.1007/978-3-031-09016-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-09015-8" title="Special:BookSources/978-3-031-09015-8"><bdi>978-3-031-09015-8</bdi></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:244921148">244921148</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multi-Winner+Voting+with+Approval+Preferences&rft.series=SpringerBriefs+in+Intelligent+Systems&rft.date=2023&rft_id=info%3Aarxiv%2F2007.01795&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A244921148%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-031-09016-5&rft.isbn=978-3-031-09015-8&rft.aulast=Lackner&rft.aufirst=Martin&rft.au=Skowron%2C+Piotr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProportional+approval+voting" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPiotr_Faliszewski,_Piotr_Skowron,_Arkadii_Slinko,_Nimrod_Talmon2017" class="citation book cs1">Piotr Faliszewski, Piotr Skowron, Arkadii Slinko, Nimrod Talmon (2017-10-26). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0qY8DwAAQBAJ&dq=multiwinner++voting+a+new+challenge&pg=PA27">"Multiwinner Voting: A New Challenge for Social Choice Theory"</a>. In Endriss, Ulle (ed.). <i>Trends in Computational Social Choice</i>. AI Access. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-326-91209-3" title="Special:BookSources/978-1-326-91209-3"><bdi>978-1-326-91209-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Multiwinner+Voting%3A+A+New+Challenge+for+Social+Choice+Theory&rft.btitle=Trends+in+Computational+Social+Choice&rft.pub=AI+Access&rft.date=2017-10-26&rft.isbn=978-1-326-91209-3&rft.au=Piotr+Faliszewski%2C+Piotr+Skowron%2C+Arkadii+Slinko%2C+Nimrod+Talmon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0qY8DwAAQBAJ%26dq%3Dmultiwinner%2B%2Bvoting%2Ba%2Bnew%2Bchallenge%26pg%3DPA27&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProportional+approval+voting" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li> <li><a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt method</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/Phragmen%27s_voting_rules" title="Phragmen's voting rules">Phragmen's voting rules</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-dhondt-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-dhondt_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dhondt_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-dhondt_1-2"><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="CITEREFBrillLaslierSkowron2018" class="citation journal cs1">Brill, Markus; Laslier, Jean-François; Skowron, Piotr (2018). 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Batko, Pawel; Skowron, Piotr; Faliszewski, Piotr (2021). <a rel="nofollow" class="external text" href="https://ojs.aaai.org/index.php/AAAI/article/view/16686">"An Analysis of Approval-Based Committee Rules for 2D-Euclidean Elections"</a>. <i>Proceedings of the AAAI Conference on Artificial Intelligence</i>. <b>35</b> (6): 5448–5455. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1609%2Faaai.v35i6.16686">10.1609/aaai.v35i6.16686</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:235306592">235306592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+AAAI+Conference+on+Artificial+Intelligence&rft.atitle=An+Analysis+of+Approval-Based+Committee+Rules+for+2D-Euclidean+Elections&rft.volume=35&rft.issue=6&rft.pages=5448-5455&rft.date=2021&rft_id=info%3Adoi%2F10.1609%2Faaai.v35i6.16686&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A235306592%23id-name%3DS2CID&rft.aulast=Godziszewski&rft.aufirst=Michal&rft.au=Batko%2C+Pawel&rft.au=Skowron%2C+Piotr&rft.au=Faliszewski%2C+Piotr&rft_id=https%3A%2F%2Fojs.aaai.org%2Findex.php%2FAAAI%2Farticle%2Fview%2F16686&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProportional+approval+voting" class="Z3988"></span></span> </li> <li id="cite_note-pav-approx2-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-pav-approx2_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pav-approx2_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudyczManurangsiMarcinkowskiSornat2020" class="citation book cs1">Dudycz, Szymon; Manurangsi, Pasin; Marcinkowski, Jan; Sornat, Krzysztof (2020). "Tight Approximation for Proportional Approval Voting". <i>Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence</i>. IJCAI-20. pp. 276–282. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24963%2Fijcai.2020%2F39">10.24963/ijcai.2020/39</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9992411-6-5" title="Special:BookSources/978-0-9992411-6-5"><bdi>978-0-9992411-6-5</bdi></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:220484671">220484671</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tight+Approximation+for+Proportional+Approval+Voting&rft.btitle=Proceedings+of+the+Twenty-Ninth+International+Joint+Conference+on+Artificial+Intelligence&rft.series=IJCAI-20&rft.pages=276-282&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220484671%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.24963%2Fijcai.2020%2F39&rft.isbn=978-0-9992411-6-5&rft.aulast=Dudycz&rft.aufirst=Szymon&rft.au=Manurangsi%2C+Pasin&rft.au=Marcinkowski%2C+Jan&rft.au=Sornat%2C+Krzysztof&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProportional+approval+voting" class="Z3988"></span></span> </li> <li id="cite_note-pav-approx3-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-pav-approx3_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFByrkaSkowronSornat2017" class="citation book cs1">Byrka, Jaroslaw; Skowron, Piotr; Sornat, Krzysztof (2017). <i>Proportional Approval Voting, Harmonic k-median, and Negative Association</i>. Vol. 107. pp. 26:1–26:14. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1704.02183">1704.02183</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4230%2FLIPIcs.ICALP.2018.26">10.4230/LIPIcs.ICALP.2018.26</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783959770767" title="Special:BookSources/9783959770767"><bdi>9783959770767</bdi></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:3839722">3839722</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proportional+Approval+Voting%2C+Harmonic+k-median%2C+and+Negative+Association&rft.pages=26%3A1-26%3A14&rft.date=2017&rft_id=info%3Aarxiv%2F1704.02183&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3839722%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4230%2FLIPIcs.ICALP.2018.26&rft.isbn=9783959770767&rft.aulast=Byrka&rft.aufirst=Jaroslaw&rft.au=Skowron%2C+Piotr&rft.au=Sornat%2C+Krzysztof&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProportional+approval+voting" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportional_approval_voting&action=edit&section=14" 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://pref.tools/abcvoting/">Online tool for computing PAV and other approval-based multi-winner methods</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/martinlackner/abcvoting">Python package abcvoting containing an implementation of PAV</a> (<a rel="nofollow" class="external text" href="https://pypi.org/project/abcvoting/">pypi</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 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electoral_systems_footer" title="Template:Electoral systems footer"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electoral_systems_footer" title="Template talk:Electoral systems footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electoral_systems_footer" title="Special:EditPage/Template:Electoral systems footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Electoral_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Electoral_system" title="Electoral system">Electoral systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>Part of the <a href="/wiki/Portal:Politics" title="Portal:Politics">politics</a> and <a href="/wiki/Portal:Economics" title="Portal:Economics">Economics</a> series</i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Single-winner_voting_system" class="mw-redirect" title="Single-winner voting system">Single-winner</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a> <ul><li><a href="/wiki/Combined_approval_voting" title="Combined approval voting">Combined approval voting</a></li> <li><a href="/wiki/Unified_primary" title="Unified primary">Unified primary</a></li></ul></li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a></li> <li><a href="/wiki/Bucklin_voting" title="Bucklin voting">Bucklin voting</a></li> <li><a href="/wiki/Condorcet_methods" class="mw-redirect" title="Condorcet methods">Condorcet methods</a> <ul><li><a href="/wiki/Copeland%27s_method" title="Copeland's method">Copeland's method</a></li> <li><a href="/wiki/Dodgson%27s_method" title="Dodgson's method">Dodgson's method</a></li> <li><a href="/wiki/Kemeny%E2%80%93Young_method" title="Kemeny–Young method">Kemeny–Young method</a></li> <li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax Condorcet method</a></li> <li><a href="/wiki/Nanson%27s_method" title="Nanson's method">Nanson's method</a></li> <li><a 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 class="mw-selflink selflink">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 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