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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property of electoral systems</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output 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class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Mutual_majority_criterion" title="Special:EditPage/Mutual majority criterion">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this article. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Mutual+majority+criterion%22">"Mutual majority criterion"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Mutual+majority+criterion%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Mutual+majority+criterion%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Mutual+majority+criterion%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Mutual+majority+criterion%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Mutual+majority+criterion%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">September 2010</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The <b>mutual majority criterion</b> is a criterion for evaluating <a href="/wiki/Electoral_system" title="Electoral system">electoral systems</a>. It is also known as the <b>majority criterion for solid coalitions</b> and the <b>generalized majority criterion</b>. This criterion requires that whenever a <a href="/wiki/Majority" title="Majority">majority</a> of voters prefer a group of candidates above all others, then the winner must be a candidate from that group.<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> The mutual majority criterion may also be thought of as the single-winner case of Droop-<a href="/wiki/Proportionality_for_solid_coalitions" title="Proportionality for solid coalitions">Proportionality for Solid Coalitions</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mutual_majority_criterion&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let L be a subset of candidates. A <a href="/wiki/Solid_coalition" title="Solid coalition">solid coalition</a> in support of L is a group of voters who strictly prefer all members of L to all candidates outside of L. In other words, each member of the solid coalition ranks their least-favorite member of L higher than their favorite member outside L. Note that the members of the solid coalition may rank the members of L differently. </p><p>The mutual majority criterion says that if there is a <a href="/wiki/Solid_coalition" title="Solid coalition">solid coalition</a> of voters in support of L, and this solid coalition consists of more than half of all voters, then the winner of the election must belong to L. </p> <div class="mw-heading mw-heading3"><h3 id="Relationships_to_other_criteria">Relationships to other criteria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mutual_majority_criterion&action=edit&section=2" title="Edit section: Relationships to other criteria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is similar to but stricter than the <a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">majority criterion</a>, where the requirement applies only to the case that L is only one single candidate. It is also stricter than the <a href="/wiki/Majority_loser_criterion" title="Majority loser criterion">majority loser criterion</a>, which only applies when L consists of all candidates except one.<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><p>All <a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith-efficient</a> <a href="/wiki/Condorcet_methods" class="mw-redirect" title="Condorcet methods">Condorcet methods</a> pass the mutual majority criterion.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Methods which pass mutual majority but fail the <a href="/wiki/Condorcet_criterion" class="mw-redirect" title="Condorcet criterion">Condorcet criterion</a> may nullify the voting power of voters outside the mutual majority whenever they fail to elect the Condorcet winner. </p> <div class="mw-heading mw-heading2"><h2 id="By_method">By method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mutual_majority_criterion&action=edit&section=3" title="Edit section: By method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Anti-plurality voting</a>, <a href="/wiki/Range_voting" class="mw-redirect" title="Range voting">range voting</a>, and the <a href="/wiki/Borda_count" title="Borda count">Borda count</a> fail the <a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">majority-favorite criterion</a> and hence fail the mutual majority criterion. </p><p>The <a href="/wiki/Schulze_method" title="Schulze method">Schulze method</a>, <a href="/wiki/Ranked_pairs" title="Ranked pairs">ranked pairs</a>, <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">instant-runoff voting</a>, <a href="/wiki/Nanson%27s_method" title="Nanson's method">Nanson's method</a>, and <a href="/wiki/Bucklin_voting" title="Bucklin voting">Bucklin voting</a> pass this criterion. </p> <div class="mw-heading mw-heading3"><h3 id="Borda_count">Borda count</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mutual_majority_criterion&action=edit&section=4" title="Edit section: Borda count"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i><a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">Majority criterion#Borda count</a></i></dd></dl> <p>The mutual majority criterion implies the <a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">majority criterion</a> so the Borda count's failure of the latter is also a failure of the mutual majority criterion. The set solely containing candidate A is a set S as described in the definition. </p> <div class="mw-heading mw-heading3"><h3 id="Minimax">Minimax</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mutual_majority_criterion&action=edit&section=5" title="Edit section: Minimax"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Minimax_Condorcet" class="mw-redirect" title="Minimax Condorcet">Minimax Condorcet</a></div> <p>Assume four candidates A, B, C, and D with 100 voters and the following preferences: </p> <table class="wikitable"> <tbody><tr> <th>19 voters</th> <th>17 voters</th> <th>17 voters</th> <th>16 voters</th> <th>16 voters</th> <th>15 voters </th></tr> <tr> <td>1. C</td> <td>1. D</td> <td>1. B</td> <td>1. D</td> <td>1. A</td> <td>1. D </td></tr> <tr> <td>2. A</td> <td>2. C</td> <td>2. C</td> <td>2. B</td> <td>2. B</td> <td>2. A </td></tr> <tr> <td>3. B</td> <td>3. A</td> <td>3. A</td> <td>3. C</td> <td>3. C</td> <td>3. B </td></tr> <tr> <td>4. D</td> <td>4. B</td> <td>4. D</td> <td>4. A</td> <td>4. D</td> <td>4. C </td></tr></tbody></table> <p>The results would be tabulated as follows: </p> <table class="wikitable" border="1"> <caption>Pairwise election results </caption> <tbody><tr> <td colspan="2" rowspan="2"> </td> <td colspan="4" bgcolor="#c0c0ff" align="center">X </td></tr> <tr> <td bgcolor="#c0c0ff">A </td> <td bgcolor="#c0c0ff">B </td> <td bgcolor="#c0c0ff">C </td> <td bgcolor="#c0c0ff">D </td></tr> <tr> <td bgcolor="#ffc0c0" rowspan="4">Y </td> <td bgcolor="#ffc0c0">A </td> <td> </td> <td bgcolor="#ffe0e0">[X] 33 <br />[Y] 67 </td> <td bgcolor="#e0e0ff">[X] 69 <br />[Y] 31 </td> <td bgcolor="#ffe0e0">[X] 48 <br />[Y] 52 </td></tr> <tr> <td bgcolor="#ffc0c0">B </td> <td bgcolor="#e0e0ff">[X] 67 <br />[Y] 33 </td> <td> </td> <td bgcolor="#ffe0e0">[X] 36 <br />[Y] 64 </td> <td bgcolor="#ffe0e0">[X] 48 <br />[Y] 52 </td></tr> <tr> <td bgcolor="#ffc0c0">C </td> <td bgcolor="#ffe0e0">[X] 31 <br />[Y] 69 </td> <td bgcolor="#e0e0ff">[X] 64 <br />[Y] 36 </td> <td> </td> <td bgcolor="#ffe0e0">[X] 48 <br />[Y] 52 </td></tr> <tr> <td bgcolor="#ffc0c0">D </td> <td bgcolor="#e0e0ff">[X] 52 <br />[Y] 48 </td> <td bgcolor="#e0e0ff">[X] 52 <br />[Y] 48 </td> <td bgcolor="#e0e0ff">[X] 52 <br />[Y] 48 </td> <td> </td></tr> <tr> <td colspan="2" bgcolor="#c0c0ff">Pairwise election results (won-tied-lost): </td> <td>2-0-1 </td> <td>2-0-1 </td> <td>2-0-1 </td> <td>0-0-3 </td></tr> <tr> <td colspan="2" bgcolor="#c0c0ff">worst pairwise defeat (winning votes): </td> <td>69 </td> <td>67 </td> <td>64 </td> <td>52 </td></tr> <tr> <td colspan="2" bgcolor="#c0c0ff">worst pairwise defeat (margins): </td> <td>38 </td> <td>34 </td> <td>28 </td> <td>4 </td></tr> <tr> <td colspan="2" bgcolor="#c0c0ff">worst pairwise opposition: </td> <td>69 </td> <td>67 </td> <td>64 </td> <td>52 </td></tr></tbody></table> <ul><li>[X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption</li> <li>[Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption</li></ul> <p><b>Result</b>: Candidates A, B and C each are strictly preferred by more than the half of the voters (52%) over D, so {A, B, C} is a set S as described in the definition and D is a Condorcet loser. Nevertheless, Minimax declares <b>D</b> the winner because its biggest defeat is significantly the smallest compared to the defeats A, B and C caused each other. </p> <div class="mw-heading mw-heading3"><h3 id="Plurality">Plurality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mutual_majority_criterion&action=edit&section=6" title="Edit section: Plurality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Tennessee"></span> </p> <div style="float: left;"> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tenn_voting_example" title="Template:Tenn voting example"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tenn_voting_example" title="Template talk:Tenn voting example"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tenn_voting_example" title="Special:EditPage/Template:Tenn voting example"><abbr title="Edit this template">e</abbr></a></li></ul></div></div> <p><span typeof="mw:File"><a href="/wiki/File:Tennessee_map_for_voting_example.svg" class="mw-file-description"><img alt="Tennessee and its four major cities: Memphis in the far west; Nashville in the center; Chattanooga in the east; and Knoxville in the far northeast" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Tennessee_map_for_voting_example.svg/500px-Tennessee_map_for_voting_example.svg.png" decoding="async" width="500" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Tennessee_map_for_voting_example.svg/750px-Tennessee_map_for_voting_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Tennessee_map_for_voting_example.svg/1000px-Tennessee_map_for_voting_example.svg.png 2x" data-file-width="780" data-file-height="190" /></a></span> </p><p>Suppose that <a href="/wiki/Tennessee" title="Tennessee">Tennessee</a> is holding an election on the location of its <a href="/wiki/Capital_city" title="Capital city">capital</a>. The population is concentrated around four major cities. <a href="/wiki/Spatial_voting" title="Spatial voting">All voters want the capital to be as close to them as possible.</a> The options are: </p> <ul><li><a href="/wiki/Memphis,_Tennessee" title="Memphis, Tennessee">Memphis</a>, the largest city, but far from the others (42% of voters)</li> <li><a href="/wiki/Nashville,_Tennessee" title="Nashville, Tennessee">Nashville</a>, near the center of the state (26% of voters)</li> <li><a href="/wiki/Chattanooga,_Tennessee" title="Chattanooga, Tennessee">Chattanooga</a>, somewhat east (15% of voters)</li> <li><a href="/wiki/Knoxville,_Tennessee" title="Knoxville, Tennessee">Knoxville</a>, far to the northeast (17% of voters)</li></ul> <p>The preferences of each region's voters are: </p> <table class="wikitable"> <tbody><tr> <th width="25%" style="background-color: #ffdddd">42% of voters<br /><small>Far-West</small> </th> <th width="25%" style="background-color: #ccffcc">26% of voters<br /><small>Center</small> </th> <th width="25%" style="background-color: #ddddff">15% of voters<br /><small>Center-East</small> </th> <th width="25%" style="background-color: #ffeedd">17% of voters<br /><small>Far-East</small> </th></tr> <tr> <td> <ol style="margin-left: 1.5em;"> <li> <b>Memphis</b> </li><li> Nashville </li><li> Chattanooga </li><li> Knoxville </li></ol> </td> <td> <ol style="margin-left: 1.5em;"> <li> <b>Nashville</b> </li><li> Chattanooga </li><li> Knoxville </li><li> Memphis </li></ol> </td> <td> <ol style="margin-left: 1.5em;"> <li> <b>Chattanooga</b> </li><li> Knoxville </li><li> Nashville </li><li> Memphis </li></ol> </td> <td> <ol style="margin-left: 1.5em;"> <li> <b>Knoxville</b> </li><li> Chattanooga </li><li> Nashville </li><li> Memphis </li></ol> </td></tr></tbody></table> <p><br /> 58% of the voters prefer Nashville, Chattanooga and Knoxville to Memphis. Therefore, the three eastern cities build a set <i>S</i> as described in the definition. But, since the supporters of the three cities split their votes, Memphis wins under plurality voting. </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=Mutual_majority_criterion&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Majority_winner_criterion" title="Majority winner criterion">Majority criterion</a></li> <li><a href="/wiki/Majority_loser_criterion" title="Majority loser criterion">Majority loser criterion</a></li> <li><a href="/wiki/Voting_system" class="mw-redirect" title="Voting system">Voting system</a></li> <li><a href="/wiki/Voting_system_criterion" class="mw-redirect" title="Voting system criterion">Voting system criterion</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=Mutual_majority_criterion&action=edit&section=8" 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"><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="CITEREFGreen-Armytage2004" class="citation journal cs1">Green-Armytage, James (2004). <a rel="nofollow" class="external text" href="https://www.votingmatters.org.uk/ISSUE19/i19p2.pdf">"Cardinal-weighted pairwise comparison"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Voting_matters" title="Voting matters">Voting matters</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Voting+matters&rft.atitle=Cardinal-weighted+pairwise+comparison&rft.date=2004&rft.aulast=Green-Armytage&rft.aufirst=James&rft_id=https%3A%2F%2Fwww.votingmatters.org.uk%2FISSUE19%2Fi19p2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMutual+majority+criterion" 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="CITEREFTideman2006" class="citation book cs1">Tideman, Nicolaus (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RN5q_LuByUoC"><i>Collective Decisions and Voting: The Potential for Public Choice</i></a>. Ashgate Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7546-4717-1" title="Special:BookSources/978-0-7546-4717-1"><bdi>978-0-7546-4717-1</bdi></a>. <q>Note that mutual majority consistency implies majority consistency.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collective+Decisions+and+Voting%3A+The+Potential+for+Public+Choice&rft.pub=Ashgate+Publishing&rft.date=2006&rft.isbn=978-0-7546-4717-1&rft.aulast=Tideman&rft.aufirst=Nicolaus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRN5q_LuByUoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMutual+majority+criterion" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreen-Armytage2011" class="citation magazine cs1">Green-Armytage, James (October 2011). <a rel="nofollow" class="external text" href="https://www.votingmatters.org.uk/ISSUE29/ISSUE29.pdf">"Four Condorcet-Hare Hybrid Methods for Single-Winner Elections"</a> <span class="cs1-format">(PDF)</span>. <i>Voting Matters</i>. No. 29. pp. <span class="nowrap">1–</span>14. <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:15220771">15220771</a>. <q>Meanwhile, they possess Smith consistency [efficiency], along with properties that are implied by this, such as [...] mutual majority.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Voting+Matters&rft.atitle=Four+Condorcet-Hare+Hybrid+Methods+for+Single-Winner+Elections&rft.issue=29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E14&rft.date=2011-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15220771%23id-name%3DS2CID&rft.aulast=Green-Armytage&rft.aufirst=James&rft_id=https%3A%2F%2Fwww.votingmatters.org.uk%2FISSUE29%2FISSUE29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMutual+majority+criterion" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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.navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Electoral_systems390" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electoral_systems_footer" title="Template:Electoral systems footer"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electoral_systems_footer" title="Template talk:Electoral systems footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electoral_systems_footer" title="Special:EditPage/Template:Electoral systems footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Electoral_systems390" style="font-size:114%;margin:0 4em"><a href="/wiki/Electoral_system" title="Electoral system">Electoral systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>Part of the <a href="/wiki/Portal:Politics" title="Portal:Politics">politics</a> and <a href="/wiki/Portal:Economics" title="Portal:Economics">Economics</a> series</i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Single-winner_voting_system" class="mw-redirect" title="Single-winner voting system">Single-winner</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a> <ul><li><a href="/wiki/Combined_approval_voting" title="Combined approval voting">Combined approval voting</a></li> <li><a href="/wiki/Unified_primary" title="Unified primary">Unified primary</a></li></ul></li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a></li> <li><a href="/wiki/Bucklin_voting" title="Bucklin voting">Bucklin voting</a></li> <li><a href="/wiki/Condorcet_methods" class="mw-redirect" title="Condorcet methods">Condorcet methods</a> <ul><li><a href="/wiki/Copeland%27s_method" title="Copeland's method">Copeland's method</a></li> <li><a href="/wiki/Dodgson%27s_method" title="Dodgson's method">Dodgson's method</a></li> <li><a href="/wiki/Kemeny%E2%80%93Young_method" title="Kemeny–Young method">Kemeny–Young method</a></li> <li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax Condorcet method</a></li> <li><a href="/wiki/Nanson%27s_method" title="Nanson's method">Nanson's method</a></li> <li><a href="/wiki/Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li> <li><a href="/wiki/Schulze_method" title="Schulze method">Schulze method</a></li></ul></li> <li><a href="/wiki/Exhaustive_ballot" title="Exhaustive ballot">Exhaustive ballot</a></li> <li><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First-past-the-post voting</a></li> <li><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff voting</a> <ul><li><a href="/wiki/Coombs%27_method" title="Coombs' method">Coombs' method</a></li> <li><a href="/wiki/Contingent_vote" title="Contingent vote">Contingent vote</a></li> <li><a href="/wiki/Supplementary_vote" class="mw-redirect" title="Supplementary vote">Supplementary vote</a></li></ul></li> <li><a href="/wiki/Majority_rule" title="Majority rule">Simple majoritarianism</a></li> <li><a href="/wiki/Plurality_voting_system" class="mw-redirect" title="Plurality voting system">Plurality</a></li> <li><a href="/wiki/Positional_voting_system" class="mw-redirect" title="Positional voting system">Positional voting system</a></li> <li><a href="/wiki/Score_voting" title="Score voting">Score voting</a></li> <li><a href="/wiki/STAR_voting" title="STAR voting">STAR voting</a></li> <li><a href="/wiki/Two-round_system" title="Two-round system">Two-round system</a></li> <li><a href="/wiki/Graduated_majority_judgment" title="Graduated majority judgment">Graduated majority judgment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proportional_representation" title="Proportional representation">Proportional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member</a></li> <li><a href="/wiki/Mixed_single_vote#Proportional_systems" title="Mixed single vote">Mixed single vote</a></li> <li><a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></li> <li><a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">Proportional approval voting</a></li> <li><a href="/wiki/Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban</a></li> <li><a href="/wiki/Sequential_proportional_approval_voting" title="Sequential proportional approval voting">Sequential proportional approval voting</a></li> <li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Single transferable vote</a> <ul><li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Hare%E2%80%93Clark_electoral_system" title="Hare–Clark electoral system">Hare–Clark</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li></ul></li> <li><a href="/wiki/Spare_vote" title="Spare vote">Spare vote</a></li> <li><a href="/wiki/Indirect_single_transferable_voting" title="Indirect single transferable voting">Indirect single transferable voting</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Allocation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highest_averages_method" title="Highest averages method">Highest averages method</a> <ul><li><a href="/wiki/Sainte-Lagu%C3%AB_method" title="Sainte-Laguë method">Webster/Sainte-Laguë</a></li> <li><a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt</a></li></ul></li> <li><a href="/wiki/Largest_remainders_method" class="mw-redirect" title="Largest remainders method">Largest remainders method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quotas</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Droop_quota" title="Droop quota">Droop quota</a></li> <li><a href="/wiki/Hagenbach-Bischoff_quota" class="mw-redirect" title="Hagenbach-Bischoff quota">Hagenbach-Bischoff quota</a></li> <li><a href="/wiki/Hare_quota" title="Hare quota">Hare quota</a></li> <li><a href="/wiki/Imperiali_quota" title="Imperiali quota">Imperiali quota</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Mixed</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Parallel_voting" title="Parallel voting">Parallel voting</a></li> <li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">MMP</a></li> <li><a href="/wiki/Additional_member_system" class="mw-redirect" title="Additional member system">Additional member system</a></li> <li><a href="/wiki/Alternative_vote_plus" title="Alternative vote plus">Alternative vote plus</a></li> <li><a href="/wiki/Mixed_single_vote" title="Mixed single vote">Mixed single vote</a></li> <li><a href="/wiki/Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot transferable vote</a></li> <li><a href="/wiki/Scorporo" title="Scorporo">Scorporo</a></li> <li><a href="/wiki/Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage mixed system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">Single non-transferable vote</a></li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a></li> <li><a href="/wiki/Cumulative_voting" title="Cumulative voting">Cumulative voting</a></li> <li><a href="/wiki/Satisfaction_approval_voting" title="Satisfaction approval voting">Satisfaction approval voting</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Criteria</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Condorcet winner criterion</a></li> <li><a href="/wiki/Condorcet_loser_criterion" title="Condorcet loser criterion">Condorcet loser criterion</a></li> <li><a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">Consistency criterion</a></li> <li><a href="/wiki/Independence_of_clones_criterion" title="Independence of clones criterion">Independence of clones</a></li> <li><a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">Independence of irrelevant alternatives</a></li> <li><a href="/wiki/Independence_of_Smith-dominated_alternatives" title="Independence of Smith-dominated alternatives">Independence of Smith-dominated alternatives</a></li> <li><a href="/wiki/Later-no-harm_criterion" title="Later-no-harm criterion">Later-no-harm criterion</a></li> <li><a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">Majority criterion</a></li> <li><a href="/wiki/Majority_loser_criterion" title="Majority loser criterion">Majority loser criterion</a></li> <li><a href="/wiki/Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">Monotonicity criterion</a></li> <li><a class="mw-selflink selflink">Mutual majority criterion</a></li> <li><a href="/wiki/Participation_criterion" class="mw-redirect" title="Participation criterion">Participation criterion</a></li> <li><a href="/wiki/Plurality_criterion" class="mw-redirect" title="Plurality criterion">Plurality criterion</a></li> <li><a href="/wiki/Resolvability_criterion" title="Resolvability criterion">Resolvability criterion</a></li> <li><a href="/wiki/Reversal_symmetry" class="mw-redirect" title="Reversal symmetry">Reversal symmetry</a></li> <li><a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith criterion</a></li> <li><a href="/wiki/Seats-to-votes_ratio" title="Seats-to-votes ratio">Seats-to-votes ratio</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ballot" title="Ballot">Ballot</a></li> <li><a href="/wiki/Election_threshold" class="mw-redirect" title="Election threshold">Election threshold</a></li> <li><a href="/wiki/First-preference_votes" title="First-preference votes">First-preference votes</a></li> <li><a href="/wiki/Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li> <li><a href="/wiki/Spoilt_vote" title="Spoilt vote">Spoilt vote</a></li> <li><a href="/wiki/Sortition" title="Sortition">Sortition</a></li> <li><a href="/wiki/Unseating" title="Unseating">Unseating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Comparison</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Comparison_of_electoral_systems" class="mw-redirect" title="Comparison of electoral systems">Comparison of voting systems</a></li> <li><a href="/wiki/List_of_electoral_systems_by_country" title="List of electoral systems by country">Voting systems by country</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><a href="/wiki/Portal:Politics" title="Portal:Politics">Portal</a></b> — <b><a href="/wiki/Wikipedia:WikiProject_Politics" title="Wikipedia:WikiProject Politics">Project</a></b></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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