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class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Phragmén's rules for Ranked ballots</span> </div> </a> <ul id="toc-Phragmén's_rules_for_Ranked_ballots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variants" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Variants"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Variants</span> </div> </a> <button aria-controls="toc-Variants-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Variants subsection</span> </button> <ul id="toc-Variants-sublist" class="vector-toc-list"> <li id="toc-Party_voting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Party_voting"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Party voting</span> </div> </a> <ul id="toc-Party_voting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Participatory_budgeting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Participatory_budgeting"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Participatory budgeting</span> </div> </a> <ul id="toc-Participatory_budgeting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Degressive_and_regressive_proportionality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Degressive_and_regressive_proportionality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Degressive and regressive proportionality</span> </div> </a> <ul id="toc-Degressive_and_regressive_proportionality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_Phragmen's_method_to_rank_alternatives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" 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class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Homogeneity</span> </div> </a> <ul id="toc-Homogeneity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Independence_of_unelected_candidates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independence_of_unelected_candidates"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Independence of unelected candidates</span> </div> </a> <ul id="toc-Independence_of_unelected_candidates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monotonicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Monotonicity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Monotonicity</span> </div> </a> <ul id="toc-Monotonicity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Justified_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" 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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" 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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><span class="nowrap"> </span>(<a href="/wiki/List_of_electoral_systems_by_country" title="List of electoral systems by country">By country</a>)</li></ul></div></td></tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Single-member_district" title="Single-member district">Single-winner methods</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Single vote - <a href="/wiki/Plurality_voting" title="Plurality voting">plurality</a> methods</b> <ul><li><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First preference plurality (FPP)</a></li> <li><a href="/wiki/Two-round_system" title="Two-round system">Two-round</a> (<abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> <a href="/wiki/Nonpartisan_primary" title="Nonpartisan primary">Jungle primary</a>) <ul><li><a href="/wiki/Partisan_primary" class="mw-redirect" title="Partisan primary">Partisan primary</a></li></ul></li> <li><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> Alternative vote (AV)</li> <li><abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> Ranked-choice (RCV)</li></ul></li></ul> <hr /> <p><b><a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet methods</a></b> </p> <ul><li><a href="/wiki/Tideman_alternative_method" title="Tideman alternative method">Condorcet-IRV</a></li> <li><a href="/wiki/Round-robin_voting" title="Round-robin voting">Round-robin voting</a> <ul><li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax</a></li> <li><a href="/wiki/Schulze_method" title="Schulze method">Schulze</a></li> <li><a href="/wiki/Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Positional_voting" title="Positional voting">Positional voting</a></b> </p> <ul><li><a href="/wiki/First-preference_plurality" class="mw-redirect" title="First-preference plurality">Plurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">IRV</a>)</li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Baldwin%27s_method" class="mw-redirect" title="Baldwin's method">Baldwin</a>)</li> <li><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Antiplurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Coombs_method" class="mw-redirect" title="Coombs method">Coombs</a>)</li></ul> <hr /> <p><b><a href="/wiki/Rated_voting" title="Rated voting">Cardinal voting</a></b> </p> <ul><li><a href="/wiki/Score_voting" title="Score voting">Score voting</a></li> <li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a></li> <li><a href="/wiki/Highest_median_voting_rules" title="Highest median voting rules">Majority judgment</a></li> <li><a href="/wiki/STAR_voting" title="STAR voting">STAR voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Proportional_representation" title="Proportional representation">Proportional representation</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></b> <ul><li><a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a> <ul><li><a href="/wiki/Highest_averages_method" title="Highest averages method">Highest averages</a></li> <li><a href="/wiki/Largest_remainder_method" class="mw-redirect" title="Largest remainder method">Largest remainders</a></li> <li><a href="/wiki/National_remnant" title="National remnant">National remnant</a></li> <li><a href="/wiki/Biproportional_apportionment" title="Biproportional apportionment">Biproportional</a></li></ul></li> <li><a href="/wiki/Electoral_list" title="Electoral list">List type</a> <ul><li><a href="/wiki/Closed_list" title="Closed list">Closed list</a></li> <li><a href="/wiki/Open_list" title="Open list">Open list</a></li> <li><a href="/wiki/Panachage" title="Panachage">Panachage</a></li> <li><a href="/wiki/Justified_representation" title="Justified representation">List-free PR</a></li> <li><a href="/wiki/Localized_list" title="Localized list">Localized list</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b> </p> <ul><li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Hare STV</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li> <li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul> <hr /> <p><b><a href="/wiki/Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b> </p> <ul><li><a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">Thiele's method</a></li> <li><a class="mw-selflink selflink">Phragmen's method</a></li> <li><a href="/wiki/Expanding_approvals_rule" title="Expanding approvals rule">Expanding approvals rule</a></li> <li><a href="/wiki/Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li></ul> <hr /> <p><b><a href="/wiki/Fractional_social_choice" title="Fractional social choice">Fractional social choice</a></b> </p> <ul><li><a href="/wiki/Direct_representation" title="Direct representation">Direct representation</a> <ul><li><a href="/wiki/Interactive_representation" title="Interactive representation">Interactive representation</a></li> <li><a href="/wiki/Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li></ul></li> <li><a href="/wiki/Fractional_approval_voting" title="Fractional approval voting">Fractional approval voting</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li> <li><a href="/wiki/Random_ballot" title="Random ballot">Random ballot</a></li></ul> <hr /> <p><b><a href="/wiki/Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional representation</a></b> </p> <ul><li><a href="/wiki/Cumulative_voting" title="Cumulative voting">Cumulative</a> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">SNTV</a></li></ul></li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Mixed systems</a></div><div class="sidebar-list-content mw-collapsible-content"><b>By results of combination</b> <ul><li><a href="/wiki/Mixed-member_majoritarian_representation" title="Mixed-member majoritarian representation">Mixed-member majoritarian</a></li> <li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member proportional</a></li></ul> <hr /><b>By mechanism of combination</b> <ul><li><b>Non-<a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">compensatory</a></b> <ul><li><a href="/wiki/Parallel_voting" title="Parallel voting">Parallel (superposition)</a></li> <li><a href="/wiki/Coexistence_(electoral_systems)" title="Coexistence (electoral systems)">Coexistence</a></li> <li><a href="/w/index.php?title=Conditional_electoral_system&action=edit&redlink=1" class="new" title="Conditional electoral system (page does not exist)">Conditional</a></li> <li><a href="/wiki/Majority_bonus_system" title="Majority bonus system">Fusion (majority bonus)</a></li></ul></li> <li><b><a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">Compensatory</a></b> <ul><li><a href="/w/index.php?title=Seat_linkage_mixed_system&action=edit&redlink=1" class="new" title="Seat linkage mixed system (page does not exist)">Seat linkage system</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> <a href="/wiki/Additional_member_system" class="mw-redirect" title="Additional member system">'AMS'</a></li> <li><abbr style="font-size:85%" title=""><a href="/wiki/New_Zealand_English" title="New Zealand English">NZ</a>:</abbr> <a href="/wiki/Mixed-member_proportional" class="mw-redirect" title="Mixed-member proportional">'MMP'</a></li></ul></li> <li><a href="/wiki/Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage system</a> <ul><li><a href="/wiki/Scorporo" title="Scorporo">Negative vote transfer</a></li> <li><a href="/wiki/Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot</a></li></ul></li></ul></li> <li><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Supermixed systems</a> <ul><li><a href="/wiki/Dual-member_proportional_representation" class="mw-redirect" title="Dual-member proportional representation">Dual-member proportional</a></li> <li><a href="/wiki/Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban proportional</a></li> <li><a href="/wiki/Majority_jackpot_system" title="Majority jackpot system">Majority jackpot</a></li></ul></li></ul> <hr /> <p><b>By ballot type</b> </p> <ul><li><a href="/wiki/Mixed_single_vote" title="Mixed single vote">Single vote</a> <ul><li><a href="/wiki/Double_simultaneous_vote" title="Double simultaneous vote">Double simultaneous vote</a></li></ul></li> <li><a href="/wiki/Mixed_electoral_systems" class="mw-redirect" title="Mixed electoral systems">Dual-vote</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Pathological_(mathematics)#Voting" title="Pathological (mathematics)">Paradoxes and pathologies</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Spoiler effects</b> <ul><li><a href="/wiki/Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li> <li><a href="/wiki/Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloning paradox</a></li> <li><a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Frustrated majorities paradox</a></li> <li><a href="/wiki/Center_squeeze" title="Center squeeze">Center squeeze</a></li></ul> <hr /> <p><b>Pathological response</b> </p> <ul><li><a href="/wiki/Perverse_response" class="mw-redirect" title="Perverse response">Perverse response</a></li> <li><a href="/wiki/Best-is-worst_paradox" title="Best-is-worst paradox">Best-is-worst paradox</a></li> <li><a href="/wiki/No-show_paradox" title="No-show paradox">No-show paradox</a> <ul><li><a href="/wiki/Multiple_districts_paradox" title="Multiple districts paradox">Multiple districts paradox</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Strategic_voting" title="Strategic voting">Strategic voting</a></b> </p> <ul><li><a href="/wiki/Sincere_favorite_criterion" title="Sincere favorite criterion">Lesser evil voting</a></li> <li><a href="/wiki/Strategic_voting#Exaggeration" title="Strategic voting">Exaggeration</a></li> <li><a href="/wiki/Truncation_(voting)" class="mw-redirect" title="Truncation (voting)">Truncation</a></li> <li><a href="/wiki/Turkey-raising" class="mw-redirect" title="Turkey-raising">Turkey-raising</a></li></ul> <hr /> <p><b>Paradoxes of <a href="/wiki/Majority_rule" title="Majority rule">majority rule</a></b> </p> <ul><li><a href="/wiki/Tyranny_of_the_majority" title="Tyranny of the majority">Tyranny of the majority</a></li> <li><a href="/wiki/Discursive_dilemma" title="Discursive dilemma">Discursive dilemma</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Conflicting majorities paradox</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Social_choice_theory" title="Social choice theory">Social and collective choice</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Impossibility theorems</a></b> <ul><li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's theorem</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Majority impossibility</a></li> <li><a href="/wiki/Moulin%27s_impossibility_theorem" class="mw-redirect" title="Moulin's impossibility theorem">Moulin's impossibility theorem</a></li> <li><a href="/wiki/McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li> <li><a href="/wiki/Gibbard%27s_theorem" title="Gibbard's theorem">Gibbard's theorem</a></li></ul> <hr /> <p><b>Positive results</b> </p> <ul><li><a href="/wiki/Median_voter_theorem" title="Median voter theorem">Median voter theorem</a></li> <li><a href="/wiki/Condorcet%27s_jury_theorem" title="Condorcet's jury theorem">Condorcet's jury theorem</a></li> <li><a href="/wiki/May%27s_theorem" title="May's theorem">May's theorem</a></li> <li><a href="/wiki/Arrow%27s_theorem#Minimizing" class="mw-redirect" title="Arrow's theorem">Condorcet dominance theorems</a></li> <li><a href="/w/index.php?title=Harsanyi%27s_utilitarian_theorem&action=edit&redlink=1" class="new" title="Harsanyi's utilitarian theorem (page does not exist)">Harsanyi's utilitarian theorem</a></li> <li><a href="/wiki/Vickrey-Clarke-Groves_mechanism" class="mw-redirect" title="Vickrey-Clarke-Groves mechanism">VCG mechanism</a></li> <li><a href="/wiki/Quadratic_voting" title="Quadratic voting">Quadratic voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="background: var(--background-color-interactive, #efefef); color: inherit; padding-top:0.2em;"> <div class="hlist"><ul><li><span class="nowrap"><span class="mw-image-border noviewer" typeof="mw:File"><a href="/wiki/File:A_coloured_voting_box.svg" class="mw-file-description"><img alt="icon" 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template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Phragmén's voting rules</b> are rules for <a href="/wiki/Multiwinner_voting" title="Multiwinner voting">multiwinner voting</a>. They allow voters to vote for individual candidates rather than parties, but still guarantee <a href="/wiki/Proportional_representation" title="Proportional representation">proportional representation</a>. They were published by <a href="/wiki/Lars_Edvard_Phragm%C3%A9n" title="Lars Edvard Phragmén">Lars Edvard Phragmén</a> in French and Swedish between 1893 and 1899,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and translated to English by <a href="/wiki/Svante_Janson" title="Svante Janson">Svante Janson</a> in 2016.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In multiwinner approval voting, each voter can vote for one or more candidates, and the goal is to select a fixed number <i>k</i> of winners (where <i>k</i> may be, for example, the number of parliament members). The question is how to determine the set of winners? </p> <ul><li>The simplest method is <i><a href="/wiki/Multiple_non-transferable_vote" class="mw-redirect" title="Multiple non-transferable vote">multiple non-transferable vote</a></i>, in which the <i>k</i> candidates with the largest number of approvals are elected. But this method tends to select <i>k</i> candidates of the largest party, leaving the smaller parties with no representation at all.</li> <li>In the 19th century, there was much discussion regarding election systems that could guarantee <a href="/wiki/Proportional_representation" title="Proportional representation">proportional representation</a>. One solution, advocated for example by <a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt</a> in 1878, was to vote for <a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">party-lists</a> rather than individual candidates. This solution is still very common today.</li></ul> <p>Phragmén wanted to keep the vote for individual candidates, so that voters can approve candidates based on their personal merits. In the special case in which each voter approves all and only the candidates of a single party, Phragmén's methods give the same results as D'Hondt's method.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Sec.11">: Sec.11 </span></sup> However, Phragmén's method can handle more general situations, in which voters may vote for candidates from different parties (in fact, the method ignores the information on which candidate belongs to which party). </p> <div class="mw-heading mw-heading2"><h2 id="Phragmén's_rules_for_approval_ballots"><span id="Phragm.C3.A9n.27s_rules_for_approval_ballots"></span>Phragmén's rules for approval ballots</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=2" title="Edit section: Phragmén's rules for approval ballots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phragmén's method for unordered (approval) ballots can be presented in several equivalent ways.<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Sec.3">: Sec.3 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Load_balancing">Load balancing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=3" title="Edit section: Load balancing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each elected candidate creates a "load" of 1 unit. The load of a candidate must be born by voters who support him. The goal is to find a committee for which the load can be divided among the voters in the most "balanced" way. </p><p>Depending on the exact definition of "balanced" several rules are possible:<sup id="cite_ref-:2_3-0" class="reference"><a href="#cite_note-:2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li><b>Leximax-Phragmen:</b> Minimizing the maximum load, and subject to that the second-maximum load, etc. (using <a href="/wiki/Lexicographic_max-min_optimization" title="Lexicographic max-min optimization">lexicographic max-min optimization</a>).</li> <li><b>Leximin-Phragmen</b>: Maximizing the minimum load, and subject to that the second-minimum load, etc..</li> <li><b>var-Phragmen</b> or <b>Ebert's method</b>: Minimizing the <a href="/wiki/Variance" title="Variance">variance</a> of the load.</li></ul> <p>Each of these variants has two sub-variants: </p> <ul><li>A <b>global optimization</b> variant, which is usually NP-hard to compute;</li> <li>A <b>sequential</b> variant, in which candidates are selected sequentially, and in each turn, the next elected candidate is the one who attains the optimal measure among all candidates (i.e., a <a href="/wiki/Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a>).</li></ul> <p>Phragmen's original method is the sequential method that minimizes the maximum load, which is currently known as <b>Seq-Phragmen</b>.<sup id="cite_ref-:2_3-1" class="reference"><a href="#cite_note-:2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In practice, the rules that have the best axiomatic guarantees in the global-optimization category are leximax-Phragmen and var-Phragmen. Among the sequential variants, the best guarantees are given by Seq-Phragmen. </p><p>Phragmen illustrated his method by representing each voter as a vessel. The already-elected candidates are represented by water in the vessels. To elect another candidate, 1 liter of water has to be poured into the vessels corresponding to voters who voter for that candidate. The water should be distributed such that the maximum height of the water is as small as possible. </p> <div class="mw-heading mw-heading3"><h3 id="Virtual_money">Virtual money</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=4" title="Edit section: Virtual money"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seq-Phragmen can alternatively be described as the following continuous process: </p> <ul><li>Each voter starts with 0 virtual money, and receives money in a constant rate of 1 per day.</li> <li>At each time <i>t</i>, we define a not-yet-elected candidate <i>x</i> as <i>affordable</i> if the total money held by voters who approve <i>x</i> is at least 1.</li> <li>At the first time in which some candidate is affordable, we choose one affordable candidate <i>y</i> arbitrarily. We add <i>y</i> to the committee, and reset the virtual money of voters who approve <i>y</i> (as they have now "used" their virtual money to fund <i>y</i>).</li> <li>Voters keep earning virtual money and funding candidates until all <i>k</i> committee members are elected.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following simple example resembles party-list voting. There are k=6 seats and 9 candidates, denoted a,b,c,d,e,f,g,h,i. There are 63 voters with the following preferences: 31 voters approve a,b,c; 21 voters approve d,e,f; and 11 voters approve g,h,i. </p> <ul><li>Voters start earning money at a fixed rate of 1 per day. After 1/31 day (~0.0323 day), the 31 <i>abc</i> voters have 0.0323 each, so together they can fund one of their approved candidates. One of a,b,c is chosen arbitrarily; suppose it is a.</li> <li>After 1/21 day (~0.0476 day), the 31 <i>abc</i> voters have only ~0.015 each, but the 21 <i>def</i> voters have 0.0476 each, so together they can fund one of their approved candidates. One of d,e,f is chosen arbitrarily; suppose it is d.</li> <li>After ~0.0646 day, the <i>abc</i> voters again have 0.0323 each, so they buy another one of their approved candidates, say b.</li> <li>After 1/11 day (~0.091 day) the <i>ghi</i> voters have 0.091 each, so together they can fund one of their approved candidates, say g (at this point, the <i>abc</i> voters have only 0.0264 each and the <i>def</i> voters have 0.0434 each, so none of them can by another candidate).</li> <li>After 0.0952 day, the <i>def</i> voters again have 0.0476 each, so they can buy another candidate, say e.</li> <li>After 0.0969 day, the abc voters again have 0.0323 each, so they can buy another candidate, say c.</li></ul> <p>The final committee is a,b,c; d,e; g. Note that each "party" is represented approximately in proportion to its size: 3 candidates for 31 voters, 2 candidates for 21 voters, and 1 candidate for 11 voters. </p><p>Here is a more complex example. There are <i>k</i>=3 seats and 6 candidates, denoted by A, B, C, P, Q, R. The ballots are: 1034 vote for ABC, 519 vote for PQR, 90 vote for ABQ, 47 vote for APQ. The winners are elected sequentially as follows: </p> <ul><li>First, we compute for each candidate the required value of <i>t</i> so that the candidate can get a total voting-power of 1. This value is 1/1171 for A (since A appears in 1171 ballots); 1/1124 for B; 1/1034 for C; 1/566 for P; 1/656 for Q; 1/519 for R. Thus, A is elected first.</li> <li>Now, we re-compute for each candidate the required value of <i>t</i> so that the candidate can get a total voting-power of 1, keeping in mind to deduct 1/1171 from each voter who approved A. The required value for B is 1/1124+1/1171, since there are 1124 voters who approve B, and all of them already approved A. Similarly, the required value for C is 1/1034+1/1171; for Q it is 1/656+(137/656)/1171, since 137 out of 656 voters for Q already voted for A; for P it is 1/566+(47/566)/1171; and for R it is 1/519. The value is smallest for Q, so it is elected as the second winner.</li> <li>Similarly, B is elected as the third winner.</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=Phragmen%27s_voting_rules&action=edit&section=6" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Var-Phragmen and Leximax-Phragmen are NP-hard to compute, even when each agent approves 2 candidates and each candidate is approved by 3 voters. The proof is by reduction from <a href="/wiki/Maximum_independent_set" class="mw-redirect" title="Maximum independent set">Maximum independent set</a> on <a href="/wiki/Cubic_graph" title="Cubic graph">cubic graphs</a>.<sup id="cite_ref-:2_3-2" class="reference"><a href="#cite_note-:2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Leximax-Phragmen can be computed by a sequence of at most 2<i>n</i> <a href="/wiki/Mixed_integer_linear_programming" class="mw-redirect" title="Mixed integer linear programming">mixed-integer linear programs</a> with O(<i>n m</i> + <i>n</i><sup>2</sup>) variables each (where <i>n</i> is the number of voters and <i>m</i> the number of candidates); see <a href="/wiki/Lexicographic_max-min_optimization" title="Lexicographic max-min optimization">Lexicographic max-min optimization</a>. </p><p>Var-Phragmen can be computed by solving one mixed-integer <a href="/wiki/Quadratic_program" class="mw-redirect" title="Quadratic program">quadratic program</a> with O(<i>n m</i>) variables. </p><p>Seq-Phragmen can be computed in polynomial time. A naive computation shows that the run-time is O(<i>k m n</i>): there are <i>k</i> steps (one for each elected candidate); in each step, we have to check all candidates to see which of them can be funded; and for each candidate, we have to check all voters to see which of them can fund it. However, to be accurate, we need to work with rational numbers, and their magnitude grow up to <i>k</i> log <i>n</i>. Since computations in <i>b</i> bits may require O(<i>b</i><sup>2</sup>) time, the total run-time is O(<i>k<sup>3</sup> m n</i> log<sup>2</sup> n). </p> <div class="mw-heading mw-heading2"><h2 id="Phragmén's_rules_for_Ranked_ballots"><span id="Phragm.C3.A9n.27s_rules_for_Ranked_ballots"></span>Phragmén's rules for Ranked ballots</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=7" title="Edit section: Phragmén's rules for Ranked ballots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phragmén rules are commonly used with <a href="/wiki/Approval_ballot" title="Approval ballot">approval ballots</a> (that is, <a href="/wiki/Multiwinner_approval_voting" title="Multiwinner approval voting">multiwinner approval voting</a>), but they have variants using <a href="/wiki/Ranked_ballots" class="mw-redirect" title="Ranked ballots">ranked ballots</a> (that is, multiwinner <a href="/wiki/Ranked_voting" title="Ranked voting">ranked voting</a>). An adaptation for Seq-Phragmen was proposed in 1913 by a Royal Commission on the Proportional Election Method. The method has been used in Swedish elections for the distribution of seats within parties since 1921.<sup id="cite_ref-:0_2-3" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Sec.9">: Sec.9 </span></sup> </p><p>In the adapted version, in each round, each voter effectively votes only for the highest-ranked remaining candidate. Again, when a candidate is elected, his "load" of 1 unit should be distributed among the candidates who vote for him (i.e., rank him first); the load division should minimize the maximum load of a voter. </p> <div class="mw-heading mw-heading2"><h2 id="Variants">Variants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=8" title="Edit section: Variants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Party_voting">Party voting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=9" title="Edit section: Party voting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to use Phragmen's method for parties. Each voter can approve one or more parties. The procedure is the same as before, except that now, each party can be selected several times - between 0 and the total number of candidates in the party.<sup id="cite_ref-:1_4-0" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Participatory_budgeting">Participatory budgeting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=10" title="Edit section: Participatory budgeting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Seq-Phragmen rule was adapted to the more general setting of <a href="/wiki/Combinatorial_participatory_budgeting" title="Combinatorial participatory budgeting">combinatorial participatory budgeting</a>.<sup id="cite_ref-:8_5-0" class="reference"><a href="#cite_note-:8-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Degressive_and_regressive_proportionality">Degressive and regressive proportionality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=11" title="Edit section: Degressive and regressive proportionality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jaworski and Skowron<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> constructed a class of rules that generalise seq-Phragmen for degressive and regressive proportionality. Intuitively: </p> <ul><li>Degressive proportionality is obtained by assuming that the voters who already have more representatives earn money at a slower rate than those that have fewer;</li> <li>Regressive proportionality is implemented by assuming that the candidates who are approved by more voters cost less than those that garnered fewer approvals.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Using_Phragmen's_method_to_rank_alternatives"><span id="Using_Phragmen.27s_method_to_rank_alternatives"></span>Using Phragmen's method to rank alternatives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=12" title="Edit section: Using Phragmen's method to rank alternatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sequential Phragmen method can be used not only to select a subset, but also to create a ranking of alternatives, according to the order by which they are chosen. Brill and Israel<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> extend this method to <i>dynamic rankings</i>. Motivated by online Q&A applications,<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> they assume that some candidates were already chosen, and use this information in computing the ranking. They suggest two adaptations of Phragmen's rule: </p> <ul><li>Dynamic Phragmen: at each step, loop over the sequence of already-elected candidates, and divide their "cost" among their supporters. This creates, for each user, a potential "debt" - negative balance. Computing the debts can be done in time O(<i>m n</i><sup>2</sup>), where <i>m</i> is the number of candidates and <i>n</i> the number of users. Then, users start accruing money as usual, where a user can start buying new candidates only after having paid its "debt". Users buy candidates sequentially, until the new ranking is computed. The new ranking is proportional. Computing the new sequence can be done in time O(<i>m<sup>2</sup> n</i><sup>2</sup>).</li> <li>Myopic Phragmen: the "debt" of each user is computed as in Dynamic Phragmen. Then, instead of creating a complete ranking by running Sequential Phragmen, the candidates are ranked by the amount of "debt" they will create to the users. That is: the candidates are ranked by their suitability to be elected next. The resulting ranking is not necessarily proportional (in particular, when the sequence is empty, Myopic Phragmen coincides with utilitarian <a href="/wiki/Approval_voting" title="Approval voting">approval voting</a>). Computing the new sequence can be done in time O(<i>m n</i><sup>2</sup>).</li></ul> <p>They analyze the monotonicity and fairness properties of these adaptations, both theoretically and empirically. </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=Phragmen%27s_voting_rules&action=edit&section=13" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Homogeneity">Homogeneity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=14" title="Edit section: Homogeneity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For each possible ballot <i>b</i>, let <i>v<sub>b</sub></i> be the number of voters who voted exactly <i>b</i> (for example: approved exactly the same set of candidates). Let <i>p<sub>b</sub></i> be fraction of voters who voted exactly <i>b</i> (= <i>v<sub>b</sub></i> / the total number of votes). A voting method is called <i>homogeneous</i> if it depends only on the fractions <i>p<sub>b</sub></i>. So if the numbers of votes are all multiplied by the same constant, the method returns the same outcome. Phragmén's methods are homogeneous in that sense.<sup id="cite_ref-:0_2-4" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Rem.2.1">: Rem.2.1 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Independence_of_unelected_candidates">Independence of unelected candidates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=15" title="Edit section: Independence of unelected candidates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If any number of candidates is added to a ballot, but none of them is elected (even if some of them are voted for), then the outcome does not change.<sup id="cite_ref-:0_2-5" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Sec.6">: Sec.6 </span></sup> This reduces one incentive for strategic manipulation: adding "dummy" candidates to attract votes. </p> <div class="mw-heading mw-heading3"><h3 id="Monotonicity">Monotonicity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=16" title="Edit section: Monotonicity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seq-Phragmén assign seats one-by-one, so it satisfies the <a href="/wiki/Committee_monotonicity" class="mw-redirect" title="Committee monotonicity">committee monotonicity</a> property: when more seats are added, the set of winners increases (no winner loses a seat).<sup id="cite_ref-:0_2-6" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Sec.5">: Sec.5 </span></sup> </p><p>They also satisfy several other <a href="/wiki/Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">monotonicity criteria</a>.<sup id="cite_ref-:0_2-7" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Sec.14">: Sec.14 </span></sup> </p><p><b>For Phragmén's approval-ballot method</b>: if some candidate <i>C</i> is elected, and then candidate <i>C</i> earns some approvals either from new voters who vote for <i>C</i>, or from existing voters who add <i>C</i> to their ballots, and no other changes occur, then <i>C</i> is still elected. However, this monotonicity does <i>not</i> hold for pairs of candidates, even if they always appear together. For example, it is possible that candidates C, D appear together in all ballots and get two seats, but if another ballot is added for C, D, then they get together only one seat (so one of them loses a seat).<sup id="cite_ref-:0_2-8" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Ex.14.4, 14.5">: Ex.14.4, 14.5 </span></sup> Similarly, monotonicity does <i>not</i> hold in the variant with parties: a party can get more approvals but still get fewer seats. For example:<sup id="cite_ref-:1_4-1" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Suppose there are <i>k</i>=3 seats and 3 candidates: a,b,c. The ballots are: 4 for a, 7 for b, 1 for a+b, 16 for a+c, 4 for b+c. Then the elected committee is {a,b,a}. But, if one of the b voters approves a too (so that the ballots are: 4 for a, 6 for b, 2 for a+b, 16 for a+c, 4 for b+c), then the elected committee is {a,c,b}. So party a won an approval but lost a seat.</li></ul> <p><b>For Phragmén's ranked-ballot method</b>: if some candidate <i>C</i> is elected, and then candidate <i>C</i> is promoted in some of the ballots, or earns some new votes, and no other changes occur, then <i>C</i> is still elected. However, if some other changes occur simultaneously, then <i>C</i> might lose his seat. For example, it is possible that some voters change their mind, and instead of voting for A and B, they vote for C and D, and this change causes C to lose his seat.<sup id="cite_ref-:0_2-9" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Ex.13.16">: Ex.13.16 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Justified_representation">Justified representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=17" title="Edit section: Justified representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Sequential Phragmen rule satisfies an axiom known as <a href="/wiki/Proportional_justified_representation" class="mw-redirect" title="Proportional justified representation">Proportional Justified Representation</a> (PJR).<sup id="cite_ref-:2_3-3" class="reference"><a href="#cite_note-:2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This makes it one of the only methods satisfying both PJR and monotonicity. </p><p>However, it fails a stronger axiom known as <a href="/wiki/Extended_Justified_representation" class="mw-redirect" title="Extended Justified representation">Extended Justified Representation</a> (EJR). One example is given here:<sup id="cite_ref-:2_3-4" class="reference"><a href="#cite_note-:2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>There are 14 candidates: a, b, c1, ..., c12. There are 12 seats to fill.</li> <li>There are 24 voters: two voters approve {a,b,c1}; two voters approve {a,b,c2}; 6 voters approve {c1,c2,...,c12}; 5 voters approve {c2,c3,...,c12); 9 voters approve {c3,c4,...,c12}.</li> <li>Seq-Phragmen selects c1,...,c12. It violates EJR for the four voters who approve {a,b,c1} and {a,b,c2}: this group has 2 quotas and it is 2-cohesive, but no member has 2 approved winners.</li></ul> <p>Another example is given here (for the setting of parties):<sup id="cite_ref-:7_9-0" class="reference"><a href="#cite_note-:7-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <ul><li>There are 3 candidate-parties and 10 seats to fill.</li> <li>There are 10 voters, with approval sets ab,ab,ab; ac,ac,ac,ac; bc,bc; b.</li> <li>Seq-Phragmen chooses a (at time 1/7); then b; then a,b,a,b,a,b,a,b.</li> <li>Voters 1,2,3 approve all 10 candidates, but voters 4,...,10 approve only 5 candidates. However, the group of voters 4,5,6,7,8,9 all agree on party c, so EJR requires that at least one of them should approve 6 candidates, so EJR is violated (note that PJR is not violated for that group, since all 10 candidates are approved by at least one member of the group).</li></ul> <p>Seq-Phragmen also fails a different, incompatible axiom called <a href="/wiki/Justified_representation" title="Justified representation">Perfect Representation</a> (PER). </p><p>Var-Phragmen satisies PER, but fails PJR and EJR (except for the case L=1). </p><p>Leximan-Phragmen satisfies both PJR and PER, but still fails EJR. </p> <div class="mw-heading mw-heading3"><h3 id="Consistency">Consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=18" title="Edit section: Consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phragmén's methods do not satsify the <a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">consistency criterion</a>. Moreover, they do not ignore full ballots: adding voters who vote for all candidates (and thus are totally indifferent) might affect the outcome.<sup id="cite_ref-:0_2-10" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion"><sup class="reference nowrap"><span title="Page / location: Ex.15.4, 15.6, 15.8, 15.9">: Ex.15.4, 15.6, 15.8, 15.9 </span></sup></a> </p> <div class="mw-heading mw-heading3"><h3 id="Special_cases">Special cases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=19" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When there is a single seat (<i>k</i>=1): </p> <ul><li>Phragmén's approval-ballot method reduces to <a href="/wiki/Approval_voting" title="Approval voting">approval voting</a> - it always selects the candidate with the largest number of approvals.</li> <li>Phragmén's ranked-ballot method reduces to <a href="/wiki/Plurality_voting" title="Plurality voting">plurality voting</a> - it always selects the candidate ranked first by the largest number of voters.</li></ul> <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=Phragmen%27s_voting_rules&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>More information on Phragmén's methods is available at.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>Mathematical properties of Phragmen's methods vs. Thiele's methods.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li>The methods of Enestrom and Phragmen.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Implementations_and_demonstrations">Implementations and demonstrations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=21" title="Edit section: Implementations and demonstrations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Some of Phragmén's voting rules are implemented in the Python package <a rel="nofollow" class="external text" href="https://github.com/martinlackner/abcvoting"><i>abcvoting</i></a>.</li> <li>Some of Phragmén's voting rules can be tried online on the <a rel="nofollow" class="external text" href="https://pref.tools/abcvoting/">pref.tools website</a>.</li> <li>Both the simple and complicated versions<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><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> are used in the substrate of the <a href="/wiki/Cryptocurrency" title="Cryptocurrency">cryptocurrency</a> <a href="/wiki/Polkadot_(cryptocurrency)" class="mw-redirect" title="Polkadot (cryptocurrency)">Polkadot</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=22" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Motamed, Soeteman, Rey and Endriss<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> present a <i>sequential load balancing</i> mechanism, that generalizes Phragmen's rule to participatory budgeting with multiple resources. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phragmen%27s_voting_rules&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> and the "<a href="/wiki/Eureka_(word)" title="Eureka (word)">Eureka!</a>" story - analogous to the idea of load balancing</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> <li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Single transferable vote</a></li> <li><a href="/wiki/Thiele%27s_voting_rules" title="Thiele's voting rules">Thiele'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=Phragmen%27s_voting_rules&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">1. "Om proportionella val." (Summary of a public lecture). Stockholms Dagblad, 14 March 1893. 2. "Sur une m ́ethode nouvelle pour r ́ealiser, dansles ́elections, la repr ́esentation proportionelle des partis". ̈Ofversigt avKongl. Vetenskaps-Akademiens F ̈orhandlingar 1894, N:o 3, Stockholm,133–137. 3. "Proportionella val. En valteknisk studie." Svenskasp ̈orsm ̊al 25, Lars H ̈okersbergs f ̈orlag, Stockholm, 1895. 4. "Sur la th ́eorie des ́elections multiples", ̈Ofversigt avKongl. Vetenskaps-Akademiens F ̈orhandlingar 1896, N:o 3, Stockholm,181–191. 5. "Till fr ̊agan om en proportionell valmetod." Statsvetenskaplig Tidskrift2(1899), nr 2, 297–305. <a rel="nofollow" class="external free" href="http://cts.lub.lu.se/ojs/index.php/st/article/view/1949">http://cts.lub.lu.se/ojs/index.php/st/article/view/1949</a> </span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:0_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:0_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:0_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:0_2-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-:0_2-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-:0_2-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-:0_2-10"><sup><i><b>k</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJanson2018" class="citation arxiv cs1">Janson, Svante (2018-10-12). "Phragmén's and Thiele's election methods". <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/1611.08826">1611.08826</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.HO">math.HO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Phragm%C3%A9n%27s+and+Thiele%27s+election+methods&rft.date=2018-10-12&rft_id=info%3Aarxiv%2F1611.08826&rft.aulast=Janson&rft.aufirst=Svante&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-:2-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:2_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:2_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:2_3-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrillFreemanJansonLackner2023" class="citation journal cs1">Brill, Markus; Freeman, Rupert; Janson, Svante; Lackner, Martin (2023-03-06). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10858002">"Phragmén's voting methods and justified representation"</a>. <i>Mathematical Programming</i>. <b>203</b> (<span class="nowrap">1–</span>2): <span class="nowrap">47–</span>76. <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/2102.12305">2102.12305</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/10.1007%2Fs10107-023-01926-8">10.1007/s10107-023-01926-8</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1436-4646">1436-4646</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10858002">10858002</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/38344413">38344413</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Programming&rft.atitle=Phragm%C3%A9n%27s+voting+methods+and+justified+representation&rft.volume=203&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E76&rft.date=2023-03-06&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10858002%23id-name%3DPMC&rft_id=info%3Adoi%2F10.1007%2Fs10107-023-01926-8&rft_id=info%3Aarxiv%2F2102.12305&rft_id=info%3Apmid%2F38344413&rft.issn=1436-4646&rft.aulast=Brill&rft.aufirst=Markus&rft.au=Freeman%2C+Rupert&rft.au=Janson%2C+Svante&rft.au=Lackner%2C+Martin&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10858002&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-:1-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_4-1"><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="CITEREFMoraOliver2015" class="citation journal cs1 cs1-prop-foreign-lang-source">Mora, Xavier; Oliver, Maria (2015-07-28). <a rel="nofollow" class="external text" href="https://revistes.iec.cat/index.php/BSCM/article/view/89009.001">"Eleccions mitjançant el vot d'aprovació. El mètode de Phragmén i algunes variants"</a>. <i>Butlletí de la Societat Catalana de Matemàtiques</i> (in Catalan). <b>30</b> (1): <span class="nowrap">57–</span>101. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2013-9829">2013-9829</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Butllet%C3%AD+de+la+Societat+Catalana+de+Matem%C3%A0tiques&rft.atitle=Eleccions+mitjan%C3%A7ant+el+vot+d%27aprovaci%C3%B3.+El+m%C3%A8tode+de+Phragm%C3%A9n+i+algunes+variants&rft.volume=30&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E57-%3C%2Fspan%3E101&rft.date=2015-07-28&rft.issn=2013-9829&rft.aulast=Mora&rft.aufirst=Xavier&rft.au=Oliver%2C+Maria&rft_id=https%3A%2F%2Frevistes.iec.cat%2Findex.php%2FBSCM%2Farticle%2Fview%2F89009.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-:8-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-:8_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLosChristoffGrossi2022" class="citation arxiv cs1">Los, Maaike; Christoff, Zoé; Grossi, Davide (2022). "Proportional Budget Allocations: Towards a Systematization". <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/2203.12324">2203.12324</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.GT">cs.GT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Proportional+Budget+Allocations%3A+Towards+a+Systematization&rft.date=2022&rft_id=info%3Aarxiv%2F2203.12324&rft.aulast=Los&rft.aufirst=Maaike&rft.au=Christoff%2C+Zo%C3%A9&rft.au=Grossi%2C+Davide&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaworskiSkowron2022" class="citation arxiv cs1">Jaworski, Michal; Skowron, Piotr (2022). "Phragmén Rules for Degressive and Regressive Proportionality". <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/2201.04248">2201.04248</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.GT">cs.GT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Phragm%C3%A9n+Rules+for+Degressive+and+Regressive+Proportionality&rft.date=2022&rft_id=info%3Aarxiv%2F2201.04248&rft.aulast=Jaworski&rft.aufirst=Michal&rft.au=Skowron%2C+Piotr&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsraelBrill2021" class="citation arxiv cs1">Israel, Jonas; Brill, Markus (2021). "Dynamic Proportional Rankings". <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/2105.08043">2105.08043</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.GT">cs.GT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Dynamic+Proportional+Rankings&rft.date=2021&rft_id=info%3Aarxiv%2F2105.08043&rft.aulast=Israel&rft.aufirst=Jonas&rft.au=Brill%2C+Markus&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Q&A applications such as <a rel="nofollow" class="external text" href="https://www.slido.com">slido</a>, <a rel="nofollow" class="external text" href="https://www.mentimeter.com/">mentimeter</a>, <a rel="nofollow" class="external text" href="https://pigeonholelive.com/">pigeonhole live</a> or <a rel="nofollow" class="external text" href="https://speakup.digital/">speakup</a>.</span> </li> <li id="cite_note-:7-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-:7_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChandakGoelPeters2023" class="citation arxiv cs1">Chandak, Nikhil; Goel, Shashwat; Peters, Dominik (2023). "Proportional Aggregation of Preferences for Sequential Decision Making". <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/2306.14858">2306.14858</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.GT">cs.GT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Proportional+Aggregation+of+Preferences+for+Sequential+Decision+Making&rft.date=2023&rft_id=info%3Aarxiv%2F2306.14858&rft.aulast=Chandak&rft.aufirst=Nikhil&rft.au=Goel%2C+Shashwat&rft.au=Peters%2C+Dominik&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetersSkowron2020" class="citation book cs1">Peters, Dominik; Skowron, Piotr (2020-07-13). <a rel="nofollow" class="external text" href="https://doi.org/10.1145/3391403.3399465">"Proportionality and the Limits of Welfarism"</a>. <i>Proceedings of the 21st ACM Conference on Economics and Computation</i>. EC '20. New York, NY, USA: Association for Computing Machinery. pp. <span class="nowrap">793–</span>794. <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/1911.11747">1911.11747</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/10.1145%2F3391403.3399465">10.1145/3391403.3399465</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4503-7975-5" title="Special:BookSources/978-1-4503-7975-5"><bdi>978-1-4503-7975-5</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:208291203">208291203</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Proportionality+and+the+Limits+of+Welfarism&rft.btitle=Proceedings+of+the+21st+ACM+Conference+on+Economics+and+Computation&rft.place=New+York%2C+NY%2C+USA&rft.series=EC+%2720&rft.pages=%3Cspan+class%3D%22nowrap%22%3E793-%3C%2Fspan%3E794&rft.pub=Association+for+Computing+Machinery&rft.date=2020-07-13&rft_id=info%3Aarxiv%2F1911.11747&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A208291203%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F3391403.3399465&rft.isbn=978-1-4503-7975-5&rft.aulast=Peters&rft.aufirst=Dominik&rft.au=Skowron%2C+Piotr&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%2F3391403.3399465&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJansonÖberg2017" class="citation arxiv cs1">Janson, Svante; Öberg, Anders (2017). "A piecewise contractive dynamical system and election methods". <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/1709.06398">1709.06398</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.DS">math.DS</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+piecewise+contractive+dynamical+system+and+election+methods&rft.date=2017&rft_id=info%3Aarxiv%2F1709.06398&rft.aulast=Janson&rft.aufirst=Svante&rft.au=%C3%96berg%2C+Anders&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCampsMoraSaumell2019" class="citation arxiv cs1">Camps, Rosa; Mora, Xavier; Saumell, Laia (2019). "The method of Eneström and Phragmén for parliamentary elections by means of approval voting". <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/1907.10590">1907.10590</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/econ.TH">econ.TH</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+method+of+Enestr%C3%B6m+and+Phragm%C3%A9n+for+parliamentary+elections+by+means+of+approval+voting&rft.date=2019&rft_id=info%3Aarxiv%2F1907.10590&rft.aulast=Camps&rft.aufirst=Rosa&rft.au=Mora%2C+Xavier&rft.au=Saumell%2C+Laia&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/w3f/consensus">"consensus/NPoS at master · w3f/consensus"</a>. <i>GitHub</i>. 17 October 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GitHub&rft.atitle=consensus%2FNPoS+at+master+%C2%B7+w3f%2Fconsensus&rft.date=2021-10-17&rft_id=https%3A%2F%2Fgithub.com%2Fw3f%2Fconsensus&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://aaai.org/ocs/index.php/AAAI/AAAI17/paper/download/14757/13791">https://aaai.org/ocs/index.php/AAAI/AAAI17/paper/download/14757/13791</a> <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Bare_URLs" title="Wikipedia:Bare URLs"><span title="A full citation of this PDF document is required to prevent link rot. (March 2022)">bare URL PDF</span></a></i>]</sup></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://wiki.polkadot.network/docs/learn-phragmen">"Sequential Phragmén Method · Polkadot Wiki"</a>. <i>wiki.polkadot.network</i>. 30 June 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=wiki.polkadot.network&rft.atitle=Sequential+Phragm%C3%A9n+Method+%C2%B7+Polkadot+Wiki&rft.date=2023-06-30&rft_id=https%3A%2F%2Fwiki.polkadot.network%2Fdocs%2Flearn-phragmen&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhragmen%27s+voting+rules" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMotamedSoetemanReyEndriss2022" class="citation book cs1">Motamed, Nima; Soeteman, Arie; Rey, Simon; Endriss, Ulle (2022). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-031-20614-6_19">"Participatory Budgeting with Multiple Resources"</a>. In Baumeister, Dorothea; Rothe, Jörg (eds.). <i>Multi-Agent Systems</i>. Lecture Notes in Computer Science. 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