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<li id="toc-Upper_apportionment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Upper_apportionment"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Upper apportionment</span> </div> </a> <ul id="toc-Upper_apportionment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lower_apportionment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lower_apportionment"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Lower apportionment</span> </div> </a> <ul id="toc-Lower_apportionment-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Specific_example" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Specific_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Specific example</span> </div> </a> <button 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class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9partition_biproportionnelle_des_si%C3%A8ges" title="Répartition biproportionnelle des sièges – French" lang="fr" hreflang="fr" data-title="Répartition biproportionnelle des sièges" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Attribuzione_biproporzionale" title="Attribuzione biproporzionale – Italian" lang="it" hreflang="it" data-title="Attribuzione biproporzionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Biproporcion%C3%A1lis_mand%C3%A1tumeloszt%C3%A1s" title="Biproporcionális mandátumelosztás – Hungarian" lang="hu" 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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 class="mw-selflink selflink">Biproportional</a></li></ul></li> <li><a href="/wiki/Electoral_list" title="Electoral list">List type</a> <ul><li><a href="/wiki/Closed_list" title="Closed list">Closed list</a></li> <li><a href="/wiki/Open_list" title="Open list">Open list</a></li> <li><a href="/wiki/Panachage" title="Panachage">Panachage</a></li> <li><a href="/wiki/Justified_representation" title="Justified representation">List-free PR</a></li> <li><a href="/wiki/Localized_list" title="Localized list">Localized list</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b> </p> <ul><li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Hare STV</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li> <li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul> <hr /> <p><b><a href="/wiki/Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b> </p> <ul><li><a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">Thiele's method</a></li> <li><a href="/wiki/Phragmen%27s_voting_rules" title="Phragmen's voting rules">Phragmen's method</a></li> <li><a href="/wiki/Expanding_approvals_rule" title="Expanding approvals rule">Expanding approvals rule</a></li> <li><a href="/wiki/Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li></ul> <hr /> <p><b><a href="/wiki/Fractional_social_choice" title="Fractional social choice">Fractional social choice</a></b> </p> <ul><li><a href="/wiki/Direct_representation" title="Direct representation">Direct representation</a> <ul><li><a href="/wiki/Interactive_representation" title="Interactive representation">Interactive representation</a></li> <li><a href="/wiki/Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li></ul></li> <li><a href="/wiki/Fractional_approval_voting" title="Fractional approval voting">Fractional approval voting</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li> <li><a href="/wiki/Random_ballot" title="Random ballot">Random ballot</a></li></ul> <hr /> <p><b><a href="/wiki/Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional representation</a></b> </p> <ul><li><a href="/wiki/Cumulative_voting" title="Cumulative voting">Cumulative</a> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">SNTV</a></li></ul></li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Mixed systems</a></div><div class="sidebar-list-content mw-collapsible-content"><b>By results of combination</b> <ul><li><a href="/wiki/Mixed-member_majoritarian_representation" title="Mixed-member majoritarian representation">Mixed-member majoritarian</a></li> <li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member proportional</a></li></ul> <hr /><b>By mechanism of combination</b> <ul><li><b>Non-<a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">compensatory</a></b> <ul><li><a href="/wiki/Parallel_voting" title="Parallel voting">Parallel (superposition)</a></li> <li><a href="/wiki/Coexistence_(electoral_systems)" title="Coexistence (electoral systems)">Coexistence</a></li> <li><a href="/w/index.php?title=Conditional_electoral_system&action=edit&redlink=1" class="new" title="Conditional electoral system (page does not exist)">Conditional</a></li> <li><a href="/wiki/Majority_bonus_system" title="Majority bonus system">Fusion (majority bonus)</a></li></ul></li> <li><b><a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">Compensatory</a></b> <ul><li><a href="/w/index.php?title=Seat_linkage_mixed_system&action=edit&redlink=1" class="new" title="Seat linkage mixed system (page does not exist)">Seat linkage system</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> <a href="/wiki/Additional_member_system" class="mw-redirect" title="Additional member system">'AMS'</a></li> <li><abbr style="font-size:85%" title=""><a href="/wiki/New_Zealand_English" title="New Zealand English">NZ</a>:</abbr> <a href="/wiki/Mixed-member_proportional" class="mw-redirect" title="Mixed-member proportional">'MMP'</a></li></ul></li> <li><a href="/wiki/Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage system</a> <ul><li><a href="/wiki/Scorporo" title="Scorporo">Negative vote transfer</a></li> <li><a href="/wiki/Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot</a></li></ul></li></ul></li> <li><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Supermixed systems</a> <ul><li><a href="/wiki/Dual-member_proportional_representation" class="mw-redirect" title="Dual-member proportional representation">Dual-member proportional</a></li> <li><a href="/wiki/Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban proportional</a></li> <li><a href="/wiki/Majority_jackpot_system" title="Majority jackpot system">Majority jackpot</a></li></ul></li></ul> <hr /> <p><b>By ballot type</b> </p> <ul><li><a href="/wiki/Mixed_single_vote" title="Mixed single vote">Single vote</a> <ul><li><a href="/wiki/Double_simultaneous_vote" title="Double simultaneous vote">Double simultaneous vote</a></li></ul></li> <li><a href="/wiki/Mixed_electoral_systems" class="mw-redirect" title="Mixed electoral systems">Dual-vote</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Pathological_(mathematics)#Voting" title="Pathological (mathematics)">Paradoxes and pathologies</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Spoiler effects</b> <ul><li><a href="/wiki/Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li> <li><a href="/wiki/Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloning paradox</a></li> <li><a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Frustrated majorities paradox</a></li> <li><a href="/wiki/Center_squeeze" title="Center squeeze">Center squeeze</a></li></ul> <hr /> <p><b>Pathological response</b> </p> <ul><li><a href="/wiki/Perverse_response" class="mw-redirect" title="Perverse response">Perverse response</a></li> <li><a href="/wiki/Best-is-worst_paradox" title="Best-is-worst paradox">Best-is-worst paradox</a></li> <li><a href="/wiki/No-show_paradox" title="No-show paradox">No-show paradox</a> <ul><li><a href="/wiki/Multiple_districts_paradox" title="Multiple districts paradox">Multiple districts paradox</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Strategic_voting" title="Strategic voting">Strategic voting</a></b> </p> <ul><li><a href="/wiki/Sincere_favorite_criterion" title="Sincere favorite criterion">Lesser evil voting</a></li> <li><a href="/wiki/Strategic_voting#Exaggeration" title="Strategic voting">Exaggeration</a></li> <li><a href="/wiki/Truncation_(voting)" class="mw-redirect" title="Truncation (voting)">Truncation</a></li> <li><a href="/wiki/Turkey-raising" class="mw-redirect" title="Turkey-raising">Turkey-raising</a></li></ul> <hr /> <p><b>Paradoxes of <a href="/wiki/Majority_rule" title="Majority rule">majority rule</a></b> </p> <ul><li><a href="/wiki/Tyranny_of_the_majority" title="Tyranny of the majority">Tyranny of the majority</a></li> <li><a href="/wiki/Discursive_dilemma" title="Discursive dilemma">Discursive dilemma</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Conflicting majorities paradox</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Social_choice_theory" title="Social choice theory">Social and collective choice</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Impossibility theorems</a></b> <ul><li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's theorem</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Majority impossibility</a></li> <li><a href="/wiki/Moulin%27s_impossibility_theorem" class="mw-redirect" title="Moulin's impossibility theorem">Moulin's impossibility theorem</a></li> <li><a href="/wiki/McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li> <li><a href="/wiki/Gibbard%27s_theorem" title="Gibbard's theorem">Gibbard's theorem</a></li></ul> <hr /> <p><b>Positive results</b> </p> <ul><li><a href="/wiki/Median_voter_theorem" title="Median voter theorem">Median voter theorem</a></li> <li><a href="/wiki/Condorcet%27s_jury_theorem" title="Condorcet's jury theorem">Condorcet's jury theorem</a></li> <li><a href="/wiki/May%27s_theorem" title="May's theorem">May's theorem</a></li> <li><a href="/wiki/Arrow%27s_theorem#Minimizing" class="mw-redirect" title="Arrow's theorem">Condorcet dominance theorems</a></li> <li><a href="/w/index.php?title=Harsanyi%27s_utilitarian_theorem&action=edit&redlink=1" class="new" title="Harsanyi's utilitarian theorem (page does not exist)">Harsanyi's utilitarian theorem</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="background: var(--background-color-interactive, #efefef); color: inherit; padding-top:0.2em;"> <div class="hlist"><ul><li><span class="nowrap"><span class="mw-image-border noviewer" typeof="mw:File"><a href="/wiki/File:A_coloured_voting_box.svg" class="mw-file-description"><img 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href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output 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template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Biproportional apportionment</b> is a <a href="/wiki/Proportional_representation" title="Proportional representation">proportional representation</a> method to allocate seats in proportion to two separate characteristics. That is, for two different partitions each part receives the proportional number of seats within the total number of seats. For instance, this method could give proportional results by party and by region, or by party and by gender/ethnicity, or by any other pair of characteristics. </p> <ol><li>Example: proportional by party and by region <ul><li>Each party's share of seats is proportional to its total votes.</li> <li>Each region's share of seats is proportional to its total votes <ul><li>(or this could be based on its population-size or other criteria).</li></ul></li></ul></li> <li>Then, as nearly as possible given the totals for each region and each party: <ul><li>Each <b>region's</b> seats are allocated among <b>parties</b> in proportion to that region's votes for those parties. (The region's seats go to locally popular parties.)</li> <li>Each <b>party's</b> seats are allocated among <b>regions</b> in proportion to that party's votes in those regions. (The party's seats are in regions where it is most popular.)</li></ul></li></ol> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Process">Process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=1" title="Edit section: Process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that the method is to be used to give proportional results by party and by region. </p><p>Each party nominates a candidate list for every region. The voters vote for the parties of their region (and/or for individual candidates, in an <a href="/wiki/Open_list" title="Open list">open list</a> or <a href="/wiki/Localized_list" title="Localized list">local list</a> system). </p><p>The results are calculated in two steps: </p> <dl><dd>In the so called <i>upper apportionment</i> the seats for each party (over all regions) and the seats for each region (from all parties) are determined.</dd> <dd>In the so called <i>lower apportionment</i> the seats are distributed to the regional party list respecting the results from the upper apportionment.</dd></dl> <p>This can be seen as globally adjusting the voting power of each party's voters by the minimum amount necessary so that the region-by-region results become proportional by party. </p> <div class="mw-heading mw-heading3"><h3 id="Upper_apportionment">Upper apportionment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=2" title="Edit section: Upper apportionment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the upper apportionment the seats for each party are computed with a <a href="/wiki/Highest_averages_method" title="Highest averages method">highest averages method</a> (for example the <a href="/wiki/Sainte-Lagu%C3%AB_method" title="Sainte-Laguë method">Sainte-Laguë method</a>). This determines how many of all seats each party deserves due to the total of all their votes (that is the sum of the votes for all regional lists of that party). Analogically, the same highest averages method is used to determine how many of all seats each region deserves. </p><p>Note, that the results from the upper apportionment are final results for the number of the seats of one party (and analogically for the number of the seats of one region) within the whole voting area, the lower apportionment will only determine in which particular regions the party seats are allocated. Thus, after the upper apportionment is done, the final strength of a party/region within the parliament is definite. </p> <div class="mw-heading mw-heading3"><h3 id="Lower_apportionment">Lower apportionment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=3" title="Edit section: Lower apportionment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The lower apportionment has to distribute the seats to each regional party list in a way that respects both the apportionment of seats to the party and the apportionment of seats to the regions. </p><p>The result is obtained by an iterative process. Initially, for each region a <i>regional divisor</i> is chosen using the highest averages method for the votes allocated to each regional party list in this region. For each party a <i>party divisor</i> is initialized with 1. </p><p>Effectively, the objective of the iterative process is to modify the regional divisors and party divisors so that </p> <ul><li>the number of seats in each regional party list equals the number of their votes divided by both the regional and the party divisors which is then rounded by the rounding method of the highest averages method used, and</li> <li>the sum of the seats of all regional party lists of one party equals the number of seats computed in the upper apportionment for that party, and</li> <li>the sum of the seats of all regional party lists of one region equals the number of seats computed in the upper apportionment for that region.</li></ul> <p>The following two correction steps are executed until this objective is satisfied: </p> <ul><li>modify the party divisors such that the apportionment within each party is correct with the chosen highest averages method,</li> <li>modify the regional divisors such that the apportionment within the region is correct with the chosen highest averages method.</li></ul> <p>Using the Sainte-Laguë method, this iterative process is guaranteed to terminate with appropriate seat numbers for each regional party list. </p> <div class="mw-heading mw-heading2"><h2 id="Specific_example">Specific example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=4" title="Edit section: Specific example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose there are three parties A, B and C and three regions I, II and III and that there are 20 seats are to be distributed and that the <a href="/wiki/Sainte-Lagu%C3%AB_method" title="Sainte-Laguë method">Sainte-Laguë method</a> is used. The votes for the regional party lists are as follows: </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Party</th> <th colspan="3">Region </th> <th rowspan="2">Total </th></tr> <tr> <td>I</td> <td>II</td> <td>III </td></tr> <tr> <td>A</td> <td>123</td> <td>45</td> <td>815 </td> <th>983 </th></tr> <tr> <td>B</td> <td>912</td> <td>714</td> <td>414 </td> <th>2040 </th></tr> <tr> <td>C</td> <td>312</td> <td>255</td> <td>215 </td> <th>782 </th></tr> <tr> <th>total</th> <th>1347</th> <th>1014</th> <th>1444</th> <th>3805 </th></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Upper_apportionment_2">Upper apportionment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=5" title="Edit section: Upper apportionment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the upper apportionment, the overall seat number for the parties and the regions are determined. </p><p>Since there are 3805 voters and 20 seats, there are 190 (rounded) voters per seat. Thus the results for the distribution of the party seats is: </p> <table class="wikitable"> <tbody><tr> <th>Party</th> <th>A</th> <th>B</th> <th>C </th></tr> <tr> <td>#votes</td> <td>983</td> <td>2040</td> <td>782 </td></tr> <tr> <td>#votes/divisor</td> <td>5.2</td> <td>10.7</td> <td>4.1 </td></tr> <tr> <td>#seats</td> <td>5</td> <td>11</td> <td>4 </td></tr></tbody></table> <p>Using the divisor 190, the results for the distribution of the region seats is: </p> <table class="wikitable"> <tbody><tr> <th>Region</th> <th>I</th> <th>II</th> <th>III </th></tr> <tr> <td>#votes</td> <td>1347</td> <td>1014</td> <td>1444 </td></tr> <tr> <td>#votes/divisor</td> <td>7.1</td> <td>5.3</td> <td>7.6 </td></tr> <tr> <td>#seats</td> <td>7</td> <td>5</td> <td>8 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Lower_apportionment_2">Lower apportionment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=6" title="Edit section: Lower apportionment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Initially, regional divisors have to be found to distribute the seats of each region to the regional party lists. In the tables, for each regional party list, there are two cells, the first shows the number of votes and the second the number of seats allocated. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Party</th> <th colspan="6">region </th></tr> <tr> <td colspan="2">I</td> <td colspan="2">II</td> <td colspan="2">III </td></tr> <tr> <td>A</td> <td>123</td> <td>1</td> <td>45</td> <td>0</td> <td>815</td> <td>5 </td></tr> <tr> <td>B</td> <td>912</td> <td>4</td> <td>714</td> <td>4</td> <td>414</td> <td>2 </td></tr> <tr> <td>C</td> <td>312</td> <td>2</td> <td>255</td> <td>1</td> <td>215</td> <td>1 </td></tr> <tr> <td>total</td> <td>1347</td> <td>7</td> <td>1014</td> <td>5</td> <td>1444</td> <td>8 </td></tr> <tr> <td>regional divisor</td> <td colspan="2">205</td> <td colspan="2">200</td> <td colspan="2">180 </td></tr></tbody></table> <p>Now, the party divisors are initialized with ones and the number of seats within each party is checked (that is, compared to the number computed in the upper apportionment): </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Party</th> <th colspan="6">region </th> <th colspan="2" rowspan="2">total </th> <th rowspan="2">party <p>divisor </p> </th></tr> <tr> <td colspan="2">I</td> <td colspan="2">II</td> <td colspan="2">III </td></tr> <tr> <td>A</td> <td>123</td> <td>1</td> <td>45</td> <td>0</td> <td>815</td> <td>5</td> <td>983</td> <td bgcolor="#ffbbbb">6</td> <td>1 </td></tr> <tr> <td>B</td> <td>912</td> <td>4</td> <td>714</td> <td>4</td> <td>414</td> <td>2</td> <td>2040</td> <td bgcolor="#ffbbbb">10</td> <td>1 </td></tr> <tr> <td>C</td> <td>312</td> <td>2</td> <td>255</td> <td>1</td> <td>215</td> <td>1</td> <td>782</td> <td>4</td> <td>1 </td></tr> <tr> <td>total</td> <td>1347</td> <td>7</td> <td>1014</td> <td>5</td> <td>1444</td> <td>8</td> <td>3805</td> <td>20 </td></tr> <tr> <td>regional divisor</td> <td colspan="2">205</td> <td colspan="2">200</td> <td colspan="2">180 </td></tr></tbody></table> <p>Since not all parties have the correct number of seats, a correction step has to be executed: For parties A and B, the divisors are to be adjusted. The divisor for A has to be raised and the divisor for B has to be lowered: </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Party</th> <th colspan="6">region </th> <th colspan="2" rowspan="2">total </th> <th rowspan="2">party <p>divisor </p> </th></tr> <tr> <td colspan="2">I</td> <td colspan="2">II</td> <td colspan="2">III </td></tr> <tr> <td>A</td> <td>123</td> <td>1</td> <td>45</td> <td>0</td> <td>815</td> <td>4</td> <td>983</td> <td>5</td> <td bgcolor="#dddddd">1.1 </td></tr> <tr> <td>B</td> <td>912</td> <td>5</td> <td>714</td> <td>4</td> <td>414</td> <td>2</td> <td>2040</td> <td>11</td> <td bgcolor="#dddddd">0.95 </td></tr> <tr> <td>C</td> <td>312</td> <td>2</td> <td>255</td> <td>1</td> <td>215</td> <td>1</td> <td>782</td> <td>4</td> <td>1 </td></tr> <tr> <td>total</td> <td>1347</td> <td bgcolor="#ffbbbb">8</td> <td>1014</td> <td>5</td> <td>1444</td> <td bgcolor="#ffbbbb">7</td> <td>3805</td> <td>20 </td></tr> <tr> <td>regional divisor</td> <td colspan="2">205</td> <td colspan="2">200</td> <td colspan="2">180 </td></tr></tbody></table> <p>Now, the divisors for regions I and III have to be modified. Since region I has one seat too much (8 instead of the 7 seats computed in the upper apportionment), its divisor has to be raised; in opposite, the divisor for region III has to be lowered. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Party</th> <th colspan="6">region </th> <th colspan="2" rowspan="2">total </th> <th rowspan="2">party <p>divisor </p> </th></tr> <tr> <td colspan="2">I</td> <td colspan="2">II</td> <td colspan="2">III </td></tr> <tr> <td>A</td> <td>123</td> <td>1</td> <td>45</td> <td>0</td> <td>815</td> <td>4</td> <td>983</td> <td>5</td> <td>1.1 </td></tr> <tr> <td>B</td> <td>912</td> <td>5</td> <td>714</td> <td>4</td> <td>414</td> <td>3</td> <td>2040</td> <td bgcolor="#ffbbbb">12</td> <td>0.95 </td></tr> <tr> <td>C</td> <td>312</td> <td>1</td> <td>255</td> <td>1</td> <td>215</td> <td>1</td> <td>782</td> <td bgcolor="#ffbbbb">3</td> <td>1 </td></tr> <tr> <td>total</td> <td>1347</td> <td>7</td> <td>1014</td> <td>5</td> <td>1444</td> <td>8</td> <td>3805</td> <td>20 </td></tr> <tr> <td>regional divisor</td> <td bgcolor="#dddddd" colspan="2">210</td> <td colspan="2">200</td> <td bgcolor="#dddddd" colspan="2">170 </td></tr></tbody></table> <p>Again, the divisors for the parties have to be adjusted: </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Party</th> <th colspan="6">region </th> <th colspan="2" rowspan="2">total </th> <th rowspan="2">party <p>divisor </p> </th></tr> <tr> <td colspan="2">I</td> <td colspan="2">II</td> <td colspan="2">III </td></tr> <tr> <td>A</td> <td>123</td> <td>1</td> <td>45</td> <td>0</td> <td>815</td> <td>4</td> <td>983</td> <td>5</td> <td>1.1 </td></tr> <tr> <td>B</td> <td>912</td> <td>4</td> <td>714</td> <td>4</td> <td>414</td> <td>3</td> <td>2040</td> <td>11</td> <td bgcolor="#dddddd">0.97 </td></tr> <tr> <td>C</td> <td>312</td> <td>2</td> <td>255</td> <td>1</td> <td>215</td> <td>1</td> <td>782</td> <td>4</td> <td bgcolor="#dddddd">0.98 </td></tr> <tr> <td>total</td> <td>1347</td> <td>7</td> <td>1014</td> <td>5</td> <td>1444</td> <td>8</td> <td>3805</td> <td>20 </td></tr> <tr> <td>regional divisor</td> <td colspan="2">210</td> <td colspan="2">200</td> <td colspan="2">170 </td></tr></tbody></table> <p>Now, the numbers of seats for the three parties and the three regions match the numbers computed in the upper apportionment. Thus, the iterative process is completed. </p><p>The final seat numbers are: </p> <table class="wikitable"> <tbody><tr> <th>#seats</th> <th colspan="3">region </th> <th rowspan="2">total </th></tr> <tr> <td>Party</td> <td>I</td> <td>II</td> <td>III </td></tr> <tr> <td>A</td> <td>1</td> <td>0</td> <td>4 </td> <th>5 </th></tr> <tr> <td>B</td> <td>4</td> <td>4</td> <td>3 </td> <th>11 </th></tr> <tr> <td>C</td> <td>2</td> <td>1</td> <td>1 </td> <th>4 </th></tr> <tr> <th>total</th> <th>7</th> <th>5</th> <th>8</th> <th>20 </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=7" title="Edit section: Usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A method of biproportional appointment that was proposed in 2003 by German mathematician Friedrich Pukelsheim<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> is now used for cantonal and municipal elections in some cantons of Switzerland, i.e. <a href="/wiki/Canton_of_Zurich" title="Canton of Zurich">Zurich</a> (since 2006), <a href="/wiki/Canton_of_Aargau" class="mw-redirect" title="Canton of Aargau">Aargau</a> and <a href="/wiki/Canton_of_Schaffhausen" title="Canton of Schaffhausen">Schaffhausen</a> (since 2008), <a href="/wiki/Canton_of_Nidwalden" class="mw-redirect" title="Canton of Nidwalden">Nidwalden</a>, <a href="/wiki/Canton_of_Zug" title="Canton of Zug">Zug</a> (since 2013), <a href="/wiki/Canton_of_Schwyz" title="Canton of Schwyz">Schwyz</a> (since 2015) and <a href="/wiki/Canton_of_Valais" class="mw-redirect" title="Canton of Valais">Valais</a> (since 2017). </p><p>Biproportional appointment is also used in national elections for the <a href="/wiki/National_Assembly_(Bulgaria)" title="National Assembly (Bulgaria)">Bulgarian National Assembly</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2022)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Fair_majority_voting">Fair majority voting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biproportional_apportionment&action=edit&section=8" title="Edit section: Fair majority voting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Fair majority voting</i> is a biproportional apportionment method with single-member regions called "districts", so each district has exactly one representative. It was proposed in 2008 by <a href="/wiki/Michel_Balinski" title="Michel Balinski">Michel Balinski</a> (who also invented the single-winner voting system called <a href="/wiki/Majority_judgment" title="Majority judgment">majority judgment</a>) as a way to eliminate the power of <a href="/wiki/Gerrymandering" title="Gerrymandering">gerrymandering</a>, especially in the United States.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <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=Biproportional_apportionment&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGaffkePukelsheim2008" class="citation journal cs1">Gaffke, Norbert; Pukelsheim, Friedrich (2008-09-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/S0165489608000280">"Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems"</a>. <i>Mathematical Social Sciences</i>. <b>56</b> (2): 166–184. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.mathsocsci.2008.01.004">10.1016/j.mathsocsci.2008.01.004</a>. <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/0165-4896">0165-4896</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Social+Sciences&rft.atitle=Divisor+methods+for+proportional+representation+systems%3A+An+optimization+approach+to+vector+and+matrix+apportionment+problems&rft.volume=56&rft.issue=2&rft.pages=166-184&rft.date=2008-09-01&rft_id=info%3Adoi%2F10.1016%2Fj.mathsocsci.2008.01.004&rft.issn=0165-4896&rft.aulast=Gaffke&rft.aufirst=Norbert&rft.au=Pukelsheim%2C+Friedrich&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fabs%2Fpii%2FS0165489608000280&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABiproportional+apportionment" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBalinski2008" class="citation journal cs1">Balinski, Michel (2008-02-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1080/00029890.2008.11920503">"Fair Majority Voting (or How to Eliminate Gerrymandering)"</a>. <i>The American Mathematical Monthly</i>. <b>115</b> (2): 97–113. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2008.11920503">10.1080/00029890.2008.11920503</a>. <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/0002-9890">0002-9890</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:1139441">1139441</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Fair+Majority+Voting+%28or+How+to+Eliminate+Gerrymandering%29&rft.volume=115&rft.issue=2&rft.pages=97-113&rft.date=2008-02-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1139441%23id-name%3DS2CID&rft.issn=0002-9890&rft_id=info%3Adoi%2F10.1080%2F00029890.2008.11920503&rft.aulast=Balinski&rft.aufirst=Michel&rft_id=https%3A%2F%2Fdoi.org%2F10.1080%2F00029890.2008.11920503&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABiproportional+apportionment" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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