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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Matrix vote</span></span>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">A joint <a href="Portal%3APolitics" title="Portal:Politics">Politics</a> and <a href="Portal%3AEconomics" 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="Social_choice_theory" title="Social choice theory">Social choice</a> and <a href="Electoral_system" title="Electoral system">electoral systems</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-above">
<div class="hlist"><ul><li><a href="Social_choice_theory" title="Social choice theory">Social choice</a></li><li><a href="Mechanism_design" title="Mechanism design">Mechanism design</a></li><li><a href="Comparative_politics" title="Comparative politics">Comparative politics</a></li><li><a href="Comparison_of_voting_rules" title="Comparison of voting rules">Comparison</a></li><li><a href="List_of_electoral_systems" title="List of electoral systems">List</a><span class="nowrap"> </span>(<a href="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="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="Plurality_voting" title="Plurality voting">plurality</a> methods</b>
<ul><li><a href="First-past-the-post_voting" title="First-past-the-post voting">First preference plurality (FPP)</a></li>
<li><a href="Two-round_system" title="Two-round system">Two-round</a> (<abbr style="font-size:85%" title=""><a href="American_English" title="American English">US</a>:</abbr> <a href="Nonpartisan_primary" title="Nonpartisan primary">Jungle primary</a>)
<ul><li><a href="Partisan_primary" class="mw-redirect" title="Partisan primary">Partisan primary</a></li></ul></li>
<li><a href="Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a>
<ul><li><abbr style="font-size:85%" title=""><a href="British_English" title="British English">UK</a>:</abbr> Alternative vote (AV)</li>
<li><abbr style="font-size:85%" title=""><a href="American_English" title="American English">US</a>:</abbr> Ranked-choice (RCV)</li></ul></li>
<li><a href="Party_block_voting" class="mw-redirect" title="Party block voting">Party block voting</a></li>
<li><a href="Plurality_block_voting" title="Plurality block voting">Plurality block voting</a></li></ul>
<hr>
<p><b><a href="Condorcet_method" title="Condorcet method">Condorcet methods</a></b>
</p>
<ul><li><a href="Tideman_alternative_method" title="Tideman alternative method">Condorcet-IRV</a></li>
<li><a href="Round-robin_voting" title="Round-robin voting">Round-robin voting</a>
<ul><li><a href="Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax</a></li>
<li><a href="Kemeny_method" title="Kemeny method">Kemeny</a></li>
<li><a href="Schulze_method" title="Schulze method">Schulze</a></li>
<li><a href="Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li>
<li><a href="Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li></ul></li></ul>
<hr>
<p><b><a href="Positional_voting" title="Positional voting">Positional voting</a></b>
</p>
<ul><li><a href="First-preference_plurality" class="mw-redirect" title="First-preference plurality">Plurality</a> (<abbr style="font-size:85%" title=""><a href="Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="Instant-runoff_voting" title="Instant-runoff voting">IRV</a>)</li>
<li><a href="Borda_count" title="Borda count">Borda count</a> (<abbr style="font-size:85%" title=""><a href="Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="Baldwin's_method" class="mw-redirect" title="Baldwin's method">Baldwin</a>, <abbr style="font-size:85%" title=""><a href="Highest_median_voting_rules" title="Highest median voting rules">Mdn.</a></abbr> <a href="Bucklin_voting" title="Bucklin voting">Bucklin</a>)</li>
<li><a href="Anti-plurality_voting" title="Anti-plurality voting">Antiplurality</a> (<abbr style="font-size:85%" title=""><a href="Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="Coombs_method" class="mw-redirect" title="Coombs method">Coombs</a>)</li></ul>
<hr>
<p><b><a href="Rated_voting" title="Rated voting">Cardinal voting</a></b>
</p>
<ul><li><a href="Score_voting" title="Score voting">Score voting</a></li>
<li><a href="Approval_voting" title="Approval voting">Approval voting</a></li>
<li><a href="Highest_median_voting_rules" title="Highest median voting rules">Majority judgment</a></li>
<li><a href="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="Proportional_representation" title="Proportional representation">Proportional representation</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></b>
<ul><li><a href="Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a>
<ul><li><a href="Highest_averages_method" title="Highest averages method">Highest averages</a></li>
<li><a href="Largest_remainder_method" class="mw-redirect" title="Largest remainder method">Largest remainders</a></li>
<li><a href="National_remnant" title="National remnant">National remnant</a></li>
<li><a href="Biproportional_apportionment" title="Biproportional apportionment">Biproportional</a></li></ul></li>
<li><a href="Electoral_list" title="Electoral list">List type</a>
<ul><li><a href="Closed_list" title="Closed list">Closed list</a></li>
<li><a href="Open_list" title="Open list">Open list</a></li>
<li><a href="Panachage" title="Panachage">Panachage</a></li>
<li><a href="Justified_representation" title="Justified representation">List-free PR</a></li>
<li><a href="Localized_list" title="Localized list">Localized list</a></li></ul></li></ul>
<hr>
<p><b><a href="Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b>
</p>
<ul><li><a href="Single_transferable_vote" title="Single transferable vote">Hare STV</a></li>
<li><a href="Schulze_STV" title="Schulze STV">Schulze STV</a></li>
<li><a href="CPO-STV" title="CPO-STV">CPO-STV</a></li>
<li><a href="Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul>
<hr>
<p><b><a href="Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b>
</p>
<ul><li><a href="Proportional_approval_voting" title="Proportional approval voting">Thiele's method</a></li>
<li><a href="Phragmen's_voting_rules" title="Phragmen's voting rules">Phragmen's method</a></li>
<li><a href="Expanding_approvals_rule" title="Expanding approvals rule">Expanding approvals rule</a></li>
<li><a href="Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li></ul>
<hr>
<p><b><a href="Fractional_social_choice" title="Fractional social choice">Fractional social choice</a></b>
</p>
<ul><li><a href="Direct_representation" title="Direct representation">Direct representation</a>
<ul><li><a href="Interactive_representation" title="Interactive representation">Interactive representation</a></li>
<li><a href="Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li></ul></li>
<li><a href="Fractional_approval_voting" title="Fractional approval voting">Fractional approval voting</a></li>
<li><a href="Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li>
<li><a href="Random_ballot" title="Random ballot">Random ballot</a></li></ul>
<hr>
<p><b><a href="Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional representation</a></b>
</p>
<ul><li><a href="Cumulative_voting" title="Cumulative voting">Cumulative</a>
<ul><li><a href="Single_non-transferable_vote" title="Single non-transferable vote">SNTV</a></li></ul></li>
<li><a href="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="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="Mixed-member_majoritarian_representation" title="Mixed-member majoritarian representation">Mixed-member majoritarian</a></li>
<li><a href="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="Compensation_(electoral_systems)" title="Compensation (electoral systems)">compensatory</a></b>
<ul><li><a href="Parallel_voting" title="Parallel voting">Parallel (superposition)</a></li>
<li><a href="Coexistence_(electoral_systems)" title="Coexistence (electoral systems)">Coexistence</a></li>
<li>Conditional</li>
<li><a href="Majority_bonus_system" title="Majority bonus system">Fusion (majority bonus)</a></li></ul></li>
<li><b><a href="Compensation_(electoral_systems)" title="Compensation (electoral systems)">Compensatory</a></b>
<ul><li>Seat linkage system
<ul><li><abbr style="font-size:85%" title=""><a href="British_English" title="British English">UK</a>:</abbr> <a href="Additional_member_system" class="mw-redirect" title="Additional member system">'AMS'</a></li>
<li><abbr style="font-size:85%" title=""><a href="New_Zealand_English" title="New Zealand English">NZ</a>:</abbr> <a href="Mixed-member_proportional" class="mw-redirect" title="Mixed-member proportional">'MMP'</a></li></ul></li>
<li><a href="Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage system</a>
<ul><li><a href="Scorporo" title="Scorporo">Negative vote transfer</a></li>
<li><a href="Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot</a></li></ul></li></ul></li>
<li><a href="Mixed_electoral_system" title="Mixed electoral system">Supermixed systems</a>
<ul><li><a href="Dual-member_proportional_representation" class="mw-redirect" title="Dual-member proportional representation">Dual-member proportional</a></li>
<li><a href="Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban proportional</a></li>
<li><a href="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="Mixed_single_vote" title="Mixed single vote">Single vote</a>
<ul><li><a href="Double_simultaneous_vote" title="Double simultaneous vote">Double simultaneous vote</a></li></ul></li>
<li><a href="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="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="Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li>
<li><a href="Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloning paradox</a></li>
<li><a href="Condorcet_winner_criterion" title="Condorcet winner criterion">Frustrated majorities paradox</a></li>
<li><a href="Center_squeeze" title="Center squeeze">Center squeeze</a></li></ul>
<hr>
<p><b>Pathological response</b>
</p>
<ul><li><a href="Perverse_response" class="mw-redirect" title="Perverse response">Perverse response</a></li>
<li><a href="Best-is-worst_paradox" class="mw-redirect" title="Best-is-worst paradox">Best-is-worst paradox</a></li>
<li><a href="No-show_paradox" class="mw-redirect" title="No-show paradox">No-show paradox</a></li>
<li><a href="Multiple_districts_paradox" class="mw-redirect" title="Multiple districts paradox">Multiple districts paradox</a></li></ul>
<hr>
<p><b><a href="Strategic_voting" title="Strategic voting">Strategic voting</a></b>
</p>
<ul><li><a href="Sincere_favorite_criterion" title="Sincere favorite criterion">Lesser evil voting</a></li>
<li><a href="Strategic_voting#Exaggeration" title="Strategic voting">Exaggeration</a></li>
<li><a href="Truncation_(voting)" class="mw-redirect" title="Truncation (voting)">Truncation</a></li>
<li><a href="Turkey-raising" class="mw-redirect" title="Turkey-raising">Turkey-raising</a></li>
<li><a href="Wasted_vote" title="Wasted vote">Wasted vote</a></li></ul>
<hr>
<p><b>Paradoxes of <a href="Majority_rule" title="Majority rule">majority rule</a></b>
</p>
<ul><li><a href="Tyranny_of_the_majority" title="Tyranny of the majority">Tyranny of the majority</a></li>
<li><a href="Discursive_dilemma" class="mw-redirect" title="Discursive dilemma">Discursive dilemma</a></li>
<li><a href="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="Social_choice_theory" title="Social choice theory">Social and collective choice</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="Proof_of_impossibility" title="Proof of impossibility">Impossibility theorems</a></b>
<ul><li><a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's theorem</a></li>
<li><a href="Condorcet_paradox" title="Condorcet paradox">Majority impossibility</a></li>
<li><a href="Moulin's_impossibility_theorem" class="mw-redirect" title="Moulin's impossibility theorem">Moulin's impossibility theorem</a></li>
<li><a href="McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li>
<li><a href="Gibbard's_theorem" title="Gibbard's theorem">Gibbard's theorem</a></li></ul>
<hr>
<p><b>Positive results</b>
</p>
<ul><li><a href="Median_voter_theorem" title="Median voter theorem">Median voter theorem</a></li>
<li><a href="Condorcet's_jury_theorem" title="Condorcet's jury theorem">Condorcet's jury theorem</a></li>
<li><a href="May's_theorem" title="May's theorem">May's theorem</a></li>
<li><a href="Arrow's_theorem" class="mw-redirect" title="Arrow's theorem">Condorcet dominance theorems</a></li>
<li>Harsanyi's utilitarian theorem</li>
<li><a href="Vickrey-Clarke-Groves_mechanism" class="mw-redirect" title="Vickrey-Clarke-Groves mechanism">VCG mechanism</a></li>
<li><a href="Quadratic_voting" title="Quadratic voting">Quadratic voting</a></li></ul></div></div></td>
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<p>The <b>matrix vote</b> is a voting procedure which can be used when one group of people wishes to elect a smaller number of persons, each of whom is to have a different assignment. Examples of its use are:
</p>
<ol><li>when an association elects its executive committee; and</li>
<li>when a parliament elects its all-party power-sharing <a href="Coalition_government" title="Coalition government">coalition government</a>.</li></ol>
<p>Consider the situation in which a parliament elects a government of ten ministers.
</p>
<ul><li>The <a href="Ballot" title="Ballot">ballot paper</a> is tabular, a matrix, ten by ten: on one axis are listed the ten ministerial posts to be elected, and on the other axis are the ten preferences.</li>
<li>Each MP then casts one vote (of a maximum) of ten preferences, naming the ten ministers in her order of preference, and saying at the same time in which ministerial post she wishes each of her nominees to serve. A full <a href="Ballot" title="Ballot">ballot</a> will contain one name in each column, and one name in each row: that is, she identifies, in her order of preference, ten nominees, one for each of the ten ministerial posts.</li>
<li>The count is in two stages: a <a href="Quota_Borda_system" title="Quota Borda system">Quota Borda system</a> (QBS) is used to identify the ten most popular candidates; and then a <a href="Borda_count#Modified_Borda_count" title="Borda count">Modified Borda Count</a> determines which of these successful candidates will serve in which ministry.</li></ul>
<p>The matrix vote is proportional. It is ideally suited, therefore, to the formation of power-sharing governments, especially in post-conflict scenarios, and not least because it works without any resort to party or sectarian labels.<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><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></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Designing an All-Inclusive Democracy by Peter Emerson, published by Springer Verlag, 2007, part I pages 61-85, "Designing an All-Inclusive Democracy" <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-33163-6</bdi> (print) 978-3-540-33164-3 (online)</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"><a rel="nofollow" class="external text" href="http://www.deborda.org/faq/voting-systems/a-matrix-vote.html">What is a Matrix Vote?</a></span>
</li>
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