<div dir="ltr"><div>Jameson,<br><br>Wouldn't it be possible to abuse rule 2 to cause chaos?<br><br></div>For example, I change someone from "accept" to "reject", which would trigger his elimination.  But if he is the last candidate with over 50 points, that triggers rule 2 to be skipped entirely, resurrecting (him and) one or more other candidates, which changes the winner.  IIA violation?<br></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Oct 27, 2016 at 11:27 AM, Jameson Quinn <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com" target="_blank">jameson.quinn@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">I'm renaming the system I'd called SARA to "<a href="http://wiki.electorama.com/wiki/Majority_score_voting" target="_blank">majority score voting</a>". The SARA acronym is nice, but it relies on putting the rating categories out of their natural order (which would be the much-uglier "SAAR").<div><br></div><div>I've also renamed some of the rating categories; they are now "support, assist, accept, reject" with a default of "accept". I believe that these words give a better intuitive sense of what each rating does.<br><div><br></div><div>I've done some significant work on the wiki page. In particular, I showed criteria compliances. I also went through minor spoiler, chicken dilemma, and center squeeze scenarios, showing both honest results and probable tactics. One important note is that chicken dilemma truncation tactics will definitely backfire for a subfaction that's smaller than 25% if there's any subfaction that's larger than 25% (which must be the case in a two-way CD). Since, given reasonable margins of error for polling, there will very rarely be a smaller CD faction which can be confident that it's over 25%, this is yet another reason that CD issues will never get off the ground in this system.</div><div><br></div><div>For instance, let's say that the total error (both statistical and structural) of polls is about 4%, and that a faction will not attempt CD strategy unless it believes itself to be the smaller faction but over 25%. That would mean that the smaller faction would have to be 29% and the larger one 33%; leaving just 38% for the minority threat faction. That's an awfully tight window for the scenario to work; given polling fluctuations, it would be very unlikely that a scenario would be stably inside this window for long enough for the faction to plan and execute a CD betrayal.</div><div><br></div><div>Here's the wiki page as it currently stands:</div><div><br></div><div><h1 id="m_-182058607866077284gmail-firstHeading" class="m_-182058607866077284gmail-firstHeading" style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;font-weight:normal;margin:0px 0px 0.25em;overflow:visible;padding:0px;border-bottom:1px solid rgb(170,170,170);font-size:1.8em;line-height:1.3;font-family:"linux libertine",georgia,times,serif" lang="en">Majority score voting</h1><div id="m_-182058607866077284gmail-bodyContent" class="m_-182058607866077284gmail-mw-body-content" style="line-height:1.6;font-size:0.875em;color:rgb(37,37,37);font-family:sans-serif"><div id="m_-182058607866077284gmail-contentSub" style="font-size:11.76px;line-height:1.2em;margin:0px 0px 1.4em 1em;color:rgb(84,84,84);width:auto"></div><div id="m_-182058607866077284gmail-jump-to-nav" class="m_-182058607866077284gmail-mw-jump" style="overflow:hidden;height:0px;zoom:1;margin-bottom:1.4em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#mw-head" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"></a><a href="http://wiki.electorama.com/wiki/Majority_score_voting#p-search" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"></a></div><div id="m_-182058607866077284gmail-mw-content-text" dir="ltr" class="m_-182058607866077284gmail-mw-content-ltr" style="direction:ltr" lang="en"><p style="margin:0.5em 0px;line-height:inherit">Majority score voting chooses the candidate with highest score among the serious candidates who aren't rejected by a majority. Step by step, it works as follows:</p><ol style="margin:0.3em 0px 0px 3.2em;padding:0px"><li style="margin-bottom:0.1em"><b>Voters can support, assist, accept, or reject each candidate. Default is accept. Candidates get 2 points for each percent of "support" and 1 point for each percent of "assist", for a total of 0-200 points.</b><ul style="list-style-type:disc;margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em"><i>Obviously, you should support the best candidates (perhaps a quarter of them), and reject the worst (perhaps half of them). For the rest, the candidates who are between average and good, a simple rule of thumb is to assist when you're afraid of somebody worse, and accept when you are hoping for somebody better.</i></li></ul></li><li style="margin-bottom:0.1em"><b>Eliminate any candidates rejected by over 50%, unless that leaves no candidates with over 50 points.</b><ul style="list-style-type:disc;margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em"><i>If possible, the winner shouldn't be somebody opposed by a majority. But this shouldn't end up defaulting to a candidate who couldn't at least get accepted by over 1/2 or supported by over 1/4.</i></li></ul></li><li style="margin-bottom:0.1em"><b>Highest points wins. In case of a tie, fewest rejections wins.</b><ul style="list-style-type:disc;margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em"><i>This finds the candidate with the widest and deepest support.</i></li></ul></li></ol><p style="margin:0.5em 0px;line-height:inherit">Note: Majority score voting was originally called SARA voting, an acronym for the 4 ratings voters can give. However, putting these ratings out-of-order is confusing, even if it results in a nice-sounding acronym.</p><div id="m_-182058607866077284gmail-toc" class="m_-182058607866077284gmail-toc" style="border:1px solid rgb(170,170,170);background-color:rgb(249,249,249);padding:7px;font-size:13.3px;display:table;zoom:1"><div id="m_-182058607866077284gmail-toctitle" style="direction:ltr;text-align:center"><h2 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;margin:1em 0px 0.25em;overflow:hidden;padding:0px;border:none;font-size:13.3px;display:inline;line-height:1.3">Contents</h2><span class="m_-182058607866077284gmail-toctoggle" style="font-size:12.502px"> [<a href="http://wiki.electorama.com/wiki/Majority_score_voting#" id="m_-182058607866077284gmail-togglelink" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">hide</a>] </span></div><ul style="list-style-type:none;margin:0.3em 0px;padding:0px"><li class="m_-182058607866077284gmail-toclevel-1 m_-182058607866077284gmail-tocsection-1" style="margin-bottom:0.1em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#Criteria_compliance" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"><span class="m_-182058607866077284gmail-tocnumber" style="display:table-cell;text-decoration:inherit;padding-left:0px;padding-right:0.5em">1</span><span class="m_-182058607866077284gmail-toctext" style="display:table-cell;text-decoration:inherit">Criteria compliance</span></a></li><li class="m_-182058607866077284gmail-toclevel-1 m_-182058607866077284gmail-tocsection-2" style="margin-bottom:0.1em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#An_example" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"><span class="m_-182058607866077284gmail-tocnumber" style="display:table-cell;text-decoration:inherit;padding-left:0px;padding-right:0.5em">2</span><span class="m_-182058607866077284gmail-toctext" style="display:table-cell;text-decoration:inherit">An example</span></a></li><li class="m_-182058607866077284gmail-toclevel-1 m_-182058607866077284gmail-tocsection-3" style="margin-bottom:0.1em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#Other_scenarios_explored" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"><span class="m_-182058607866077284gmail-tocnumber" style="display:table-cell;text-decoration:inherit;padding-left:0px;padding-right:0.5em">3</span><span class="m_-182058607866077284gmail-toctext" style="display:table-cell;text-decoration:inherit">Other scenarios explored</span></a><ul style="list-style-type:none;margin:0px 0px 0px 2em;padding:0px"><li class="m_-182058607866077284gmail-toclevel-2 m_-182058607866077284gmail-tocsection-4" style="margin-bottom:0.1em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#Worst_case.3F" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"><span class="m_-182058607866077284gmail-tocnumber" style="display:table-cell;text-decoration:inherit;padding-left:0px;padding-right:0.5em">3.1</span><span class="m_-182058607866077284gmail-toctext" style="display:table-cell;text-decoration:inherit">Worst case?</span></a></li></ul></li><li class="m_-182058607866077284gmail-toclevel-1 m_-182058607866077284gmail-tocsection-5" style="margin-bottom:0.1em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#As_the_first_round_of_a_two-round_system_.28.22majority_score_with_runoff.22.29" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"><span class="m_-182058607866077284gmail-tocnumber" style="display:table-cell;text-decoration:inherit;padding-left:0px;padding-right:0.5em">4</span><span class="m_-182058607866077284gmail-toctext" style="display:table-cell;text-decoration:inherit">As the first round of a two-round system ("majority score with runoff")</span></a></li><li class="m_-182058607866077284gmail-toclevel-1 m_-182058607866077284gmail-tocsection-6" style="margin-bottom:0.1em"><a href="http://wiki.electorama.com/wiki/Majority_score_voting#Relationship_to_NOTA" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank"><span class="m_-182058607866077284gmail-tocnumber" style="display:table-cell;text-decoration:inherit;padding-left:0px;padding-right:0.5em">5</span><span class="m_-182058607866077284gmail-toctext" style="display:table-cell;text-decoration:inherit">Relationship to NOTA</span></a></li></ul></div><h2 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;font-weight:normal;margin:1em 0px 0.25em;overflow:hidden;padding:0px;border-bottom:1px solid rgb(170,170,170);font-family:"linux libertine",georgia,times,serif;line-height:1.3"><span class="m_-182058607866077284gmail-mw-headline" id="m_-182058607866077284gmail-Criteria_compliance">Criteria compliance</span><span class="m_-182058607866077284gmail-mw-editsection" style="font-size:small;margin-left:1em;vertical-align:baseline;line-height:1em;display:inline-block;font-family:sans-serif"><span class="m_-182058607866077284gmail-mw-editsection-bracket">[</span><a href="http://wiki.electorama.com/w/index.php?title=Majority_score_voting&action=edit&section=1" title="Edit section: Criteria compliance" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">edit</a><span class="m_-182058607866077284gmail-mw-editsection-bracket">]</span></span></h2><p style="margin:0.5em 0px;line-height:inherit">Majority score passes the <a href="http://wiki.electorama.com/wiki/Favorite_betrayal_criterion" title="Favorite betrayal criterion" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">favorite betrayal criterion</a>, the <a href="http://wiki.electorama.com/wiki/Majority_criterion" title="Majority criterion" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">majority criterion</a>, <a href="http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Independence of irrelevant alternatives</a>, <a href="http://wiki.electorama.com/wiki/Local_independence_of_irrelevant_alternatives" title="Local independence of irrelevant alternatives" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Local independence of irrelevant alternatives</a>(under the assumption of fixed "honest" ratings for each voter for each candidate), <a href="http://wiki.electorama.com/wiki/Independence_of_clone_alternatives" title="Independence of clone alternatives" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Independence of clone alternatives</a>, <a href="http://wiki.electorama.com/wiki/Monotonicity" title="Monotonicity" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Monotonicity</a>, <a href="http://wiki.electorama.com/w/index.php?title=Polytime&action=edit&redlink=1" class="m_-182058607866077284gmail-new" title="Polytime (page does not exist)" style="text-decoration:none;color:rgb(165,88,88);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">po<wbr>lytime</a>, <a href="http://wiki.electorama.com/w/index.php?title=Resolvability&action=edit&redlink=1" class="m_-182058607866077284gmail-new" title="Resolvability (page does not exist)" style="text-decoration:none;color:rgb(165,88,88);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">resolvability</a>, O(N) <a href="http://wiki.electorama.com/w/index.php?title=Summability&action=edit&redlink=1" class="m_-182058607866077284gmail-new" title="Summability (page does not exist)" style="text-decoration:none;color:rgb(165,88,88);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">summability</a>, and the <a href="http://wiki.electorama.com/wiki/Later-no-help_criterion" title="Later-no-help criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">later-no-help criterion</a>.</p><p style="margin:0.5em 0px;line-height:inherit">There are a few criteria for which it does not pass as such, but where it passes related but weaker criteria. These include:</p><ul style="margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em">It fails the <a href="http://wiki.electorama.com/wiki/Mutual_majority_criterion" title="Mutual majority criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">mutual majority criterion</a>, but passes if the mutual majority in question unanimously rejects all candidates outside their mutually-preferred set. It also passes if the number who give the strongest candidate in the set a rating below "support" is less than three times the amount by which the overall group exceeds 50%.</li></ul><ul style="margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em">It fails the <a href="http://wiki.electorama.com/wiki/Condorcet_criterion" title="Condorcet criterion" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Condorcet criterion</a>, but for any set of voters such that an honest majority Condorcet winner exists, there always exists a strong equilibrium set of strictly semi-honest majority score ballots that elects that CW.</li></ul><ul style="margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em">It fails the <a href="http://wiki.electorama.com/wiki/Participation_criterion" title="Participation criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">participation criterion</a> but passes the <a href="http://wiki.electorama.com/wiki/Semi-honest_participation_criterion" title="Semi-honest participation criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">semi-honest participation criterion</a>.</li></ul><p style="margin:0.5em 0px;line-height:inherit">It fails the <a href="http://wiki.electorama.com/wiki/Consistency_criterion" title="Consistency criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">consistency criterion</a>, the <a href="http://wiki.electorama.com/wiki/Condorcet_loser_criterion" title="Condorcet loser criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Condorcet loser criterion</a>, <a href="http://wiki.electorama.com/wiki/Reversibility" title="Reversibility" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">reversibility</a>, the <a href="http://wiki.electorama.com/wiki/Majority_loser_criterion" title="Majority loser criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">majority loser criterion</a>, and the <a href="http://wiki.electorama.com/wiki/Later-no-harm_criterion" title="Later-no-harm criterion" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">later-no-harm criterion</a>.</p><h2 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;font-weight:normal;margin:1em 0px 0.25em;overflow:hidden;padding:0px;border-bottom:1px solid rgb(170,170,170);font-family:"linux libertine",georgia,times,serif;line-height:1.3"><span class="m_-182058607866077284gmail-mw-headline" id="m_-182058607866077284gmail-An_example">An example</span><span class="m_-182058607866077284gmail-mw-editsection" style="font-size:small;margin-left:1em;vertical-align:baseline;line-height:1em;display:inline-block;font-family:sans-serif"><span class="m_-182058607866077284gmail-mw-editsection-bracket">[</span><a href="http://wiki.electorama.com/w/index.php?title=Majority_score_voting&action=edit&section=2" title="Edit section: An example" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">edit</a><span class="m_-182058607866077284gmail-mw-editsection-bracket">]</span></span></h2><div class="m_-182058607866077284gmail-floatright" style="clear:right;float:right;margin:0px 0px 0.5em 0.5em;border:0px"><div class="m_-182058607866077284gmail-MediaTransformError" style="width:500px;height:122px;display:inline-block">Error creating thumbnail: File missing</div></div><p style="margin:0.5em 0px;line-height:inherit">Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire <a href="http://wiki.electorama.com/wiki/Electorate" title="Electorate" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">electorate</a> lives in these four cities, and that everyone wants to live as near the capital as possible.</p><p style="margin:0.5em 0px;line-height:inherit">The candidates for the capital are:</p><ul style="margin:0.3em 0px 0px 1.6em;padding:0px"><li style="margin-bottom:0.1em"><a rel="nofollow" class="m_-182058607866077284external m_-182058607866077284gmail-text" href="http://en.wikipedia.org/wiki/Memphis" style="text-decoration:none;color:rgb(102,51,102);background-color:initial;padding-right:13px" target="_blank">Memphis</a> on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities</li><li style="margin-bottom:0.1em"><a rel="nofollow" class="m_-182058607866077284external m_-182058607866077284gmail-text" href="http://en.wikipedia.org/wiki/Nashville" style="text-decoration:none;color:rgb(102,51,102);background-color:initial;padding-right:13px" target="_blank">Nashville</a> on Wikipedia, with 26% of the voters, near the center of Tennessee</li><li style="margin-bottom:0.1em"><a rel="nofollow" class="m_-182058607866077284external m_-182058607866077284gmail-text" href="http://en.wikipedia.org/wiki/Knoxville" style="text-decoration:none;color:rgb(102,51,102);background-color:initial;padding-right:13px" target="_blank">Knoxville</a> on Wikipedia, with 17% of the voters</li><li style="margin-bottom:0.1em"><a rel="nofollow" class="m_-182058607866077284external m_-182058607866077284gmail-text" href="http://en.wikipedia.org/wiki/Chattanooga" style="text-decoration:none;color:rgb(102,51,102);background-color:initial;padding-right:13px" target="_blank">Chattanooga</a> on Wikipedia, with 15% of the voters</li></ul><p style="margin:0.5em 0px;line-height:inherit">The preferences of the voters would be divided like this:</p><table class="m_-182058607866077284gmail-wikitable" style="font-size:14px;margin:1em 0px;background-color:rgb(249,249,249);border:1px solid rgb(170,170,170);border-collapse:collapse;color:black"><tbody><tr><th style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em;background-color:rgb(255,221,221);text-align:center" width="25%">42% of voters<br><small>(close to Memphis)</small></th><th style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em;background-color:rgb(204,255,204);text-align:center" width="25%">26% of voters<br><small>(close to Nashville)</small></th><th style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em;background-color:rgb(221,221,255);text-align:center" width="25%">15% of voters<br><small>(close to Chattanooga)</small></th><th style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em;background-color:rgb(255,238,221);text-align:center" width="25%">17% of voters<br><small>(close to Knoxville)</small></th></tr><tr><td style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em"><ol style="margin:0.3em 0px 0px 3.2em;padding:0px"><li style="margin-bottom:0.1em"><b>Memphis</b></li><li style="margin-bottom:0.1em">Nashville</li><li style="margin-bottom:0.1em">Chattanooga</li><li style="margin-bottom:0.1em">Knoxville</li></ol></td><td style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em"><ol style="margin:0.3em 0px 0px 3.2em;padding:0px"><li style="margin-bottom:0.1em"><b>Nashville</b></li><li style="margin-bottom:0.1em">Chattanooga</li><li style="margin-bottom:0.1em">Knoxville</li><li style="margin-bottom:0.1em">Memphis</li></ol></td><td style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em"><ol style="margin:0.3em 0px 0px 3.2em;padding:0px"><li style="margin-bottom:0.1em"><b>Chattanooga</b></li><li style="margin-bottom:0.1em">Knoxville</li><li style="margin-bottom:0.1em">Nashville</li><li style="margin-bottom:0.1em">Memphis</li></ol></td><td style="border:1px solid rgb(170,170,170);padding:0.2em 0.4em"><ol style="margin:0.3em 0px 0px 3.2em;padding:0px"><li style="margin-bottom:0.1em"><b>Knoxville</b></li><li style="margin-bottom:0.1em">Chattanooga</li><li style="margin-bottom:0.1em">Nashville</li><li style="margin-bottom:0.1em">Memphis</li></ol></td></tr></tbody></table><p style="margin:0.5em 0px;line-height:inherit">Assume voters in each city support their own city; assist any city within 100 miles; accept any city between 100 and 200 miles; and reject any city that is over 200 miles away or is the farthest city. (These assumptions can be varied substantially without changing the result, but they seem reasonable to start with.)</p><div class="m_-182058607866077284gmail-floatright" style="clear:right;float:right;margin:0px 0px 0.5em 0.5em;border:0px"><table style="font-size:14px" border="1"><tbody><tr><th>City</th><th>Support</th><th>Assist</th><th>Accept</th><th>Reject</th><th>score</th></tr><tr><th bgcolor="#fff">Cutoff<br>(must be)</th><td bgcolor="#fff"></td><td bgcolor="#fff"></td><td bgcolor="#fff"></td><td bgcolor="#fff"><50</td><td bgcolor="#fff">>50</td></tr><tr><th bgcolor="#fff">Memphis</th><td bgcolor="#fff">42</td><td bgcolor="#fff">0</td><td bgcolor="#fff">0</td><td bgcolor="#fcc">58</td><td bgcolor="#fff">(84)</td></tr><tr><th bgcolor="#fff">Nashville</th><td bgcolor="#fff">26</td><td bgcolor="#fff">0</td><td bgcolor="#fff">74</td><td bgcolor="#fff">0</td><td bgcolor="#bfb">52</td></tr><tr><th bgcolor="#fff">Chattanooga</th><td bgcolor="#fff">15</td><td bgcolor="#fff">17</td><td bgcolor="#fff">26</td><td bgcolor="#fff">42</td><td bgcolor="#fcc">47</td></tr><tr><th bgcolor="#fff">Knoxville</th><td bgcolor="#fff">17</td><td bgcolor="#fff">15</td><td bgcolor="#fff">26</td><td bgcolor="#fff">42</td><td bgcolor="#fcc">49</td></tr></tbody></table></div><p style="margin:0.5em 0px;line-height:inherit">Chattanooga and Knoxville both get under 50 points, but Nashville is above the threshold. Thus Memphis, explicitly rejected by a majority, is eliminated. Nashville wins.</p><p style="margin:0.5em 0px;line-height:inherit">If Memphis voters tried to strategize by rejecting Nashville in the above scenario, it would have no effect.</p><p style="margin:0.5em 0px;line-height:inherit">If Chattanooga and Knoxville tried to strategize by supporting each other, this has a chance of working, but Memphis could safely defend Nashville by assisting it. Since Memphis is essentially guaranteed to be solidly rejected, the Memphis voters have nothing to lose by defensively assisting Nashville like this. A mere 13% of "assist" from Memphis's 42% — that is, under a quarter of the Memphis population — would give Nashville 65 points, more than double the combined size of Chattanooga and Knoxville, safely defending it.</p><h2 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;font-weight:normal;margin:1em 0px 0.25em;overflow:hidden;padding:0px;border-bottom:1px solid rgb(170,170,170);font-family:"linux libertine",georgia,times,serif;line-height:1.3"><span class="m_-182058607866077284gmail-mw-headline" id="m_-182058607866077284gmail-Other_scenarios_explored">Other scenarios explored</span><span class="m_-182058607866077284gmail-mw-editsection" style="font-size:small;margin-left:1em;vertical-align:baseline;line-height:1em;display:inline-block;font-family:sans-serif"><span class="m_-182058607866077284gmail-mw-editsection-bracket">[</span><a href="http://wiki.electorama.com/w/index.php?title=Majority_score_voting&action=edit&section=3" title="Edit section: Other scenarios explored" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">edit</a><span class="m_-182058607866077284gmail-mw-editsection-bracket">]</span></span></h2><p style="margin:0.5em 0px;line-height:inherit">In all of the scenarios that follow, <a href="http://wiki.electorama.com/wiki/Plurality_voting" title="Plurality voting" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">plurality voting</a> would get the wrong answer if voters vote honestly, and Majority Score would get the right answer.</p><p style="margin:0.5em 0px;line-height:inherit">Say there are two major and several minor candidates (where "major" and "minor" are defined by true support, not simply by media coverage). Voters should reject one major candidate and any minor candidates who are worse; support their favorite, whether they are major or minor; and accept everyone else. In that case, one of the two majors will be rejected by a majority; the other will win; and all minor candidates will show their true support.</p><p style="margin:0.5em 0px;line-height:inherit">Now suppose that there are two overall ideological "sides", and the majority "side" is split into two subfactions, each smaller than the full minority "side". This is called the "<a href="http://wiki.electorama.com/wiki/Chicken_dilemma" title="Chicken dilemma" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">chicken dilemma</a>", because many voting systems encourage a "game of chicken" between the two majority subfactions; whichever subfaction cooperates less will win, unless cooperation breaks down entirely and the minority wins. A very closely-balanced chicken dilemma would have 27% for subfaction A, 26% for subfaction B, and 47% for the minority Z; whereas a more typical example might have 31%, 24%, and 45%. In Majority Score, each subfaction should support their candidate, accept the other subfaction's, and reject the minority; while minority voters should support their candidate and reject both subfactions. In this case, the bigger subfaction will win.</p><p style="margin:0.5em 0px;line-height:inherit">Note that A, the bigger of the two subfactions, must necessarily be above the 50-point threshold in step 2, so Z will be safely eliminated. In many cases, B, the smaller subfaction, will fall short of 50 points. This helps explain the rationale for putting that threshold at 50 points.</p><p style="margin:0.5em 0px;line-height:inherit">This is not to say that majority score solves the chicken dilemma 100%. If subfaction B (the smaller one) is larger than 25% of the total, it is still possible for them to win if they largely reject A while the A voters largely accept B. And if both factions largely reject each other, Z can win. But the key word there is "largely"; unlike the case with <a href="http://wiki.electorama.com/wiki/Approval_voting" title="Approval voting" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">approval voting</a> or <a href="http://wiki.electorama.com/wiki/Score_voting" title="Score voting" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">score voting</a>, this strategy will not work if it's done by only a few individual voters, but only if one subfaction uses it significantly more often than the other. Also, the smaller subfaction must be larger than 25%, which is rare. Given that organizing such a coordinated betrayal in secret would be hard; that doing so openly would invite mutually-destructive retaliation; and that even if it could be done secretly and unilaterally, it would risk electing Z if the betraying subfaction did not reach 25%; it seems that majority score voting has a good chance of avoiding this problem.</p><p style="margin:0.5em 0px;line-height:inherit">As a final tricky scenario, consider what happens in the above case if the minority Z prefers the smaller subfaction B. That results in a <a href="http://wiki.electorama.com/wiki/Center_squeeze" title="Center squeeze" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">center squeeze</a> scenario: one where candidate B is a <a href="http://wiki.electorama.com/wiki/Condorcet_winner" title="Condorcet winner" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">Condorcet winner</a> (able to beat either rival in a one-on-one race) but an honest plurality loser (with the smallest faction of direct supporters). This is more common than one might think; candidate B is fighting an ideological battle on two fronts, while candidates A and C are free to triangulate towards the middle without losing supporters, so the fact that B has the smallest direct support may not reflect a lack of quality. Thus, generally speaking, as long as each faction considers their second choice to be more than half as good as their first choice (taking the worst choice as 0), it's pretty clear that the Condorcet-winner candidate B is the one who democratically "should" win this election.</p><p style="margin:0.5em 0px;line-height:inherit">In this scenario, there are two ways that B could win. B voters could reject both A and C; or A and/or C voters could assist B. Either of these makes strategic sense for the voters in question, and either or both could lead to a <a href="http://wiki.electorama.com/wiki/Strong_equilibrium" title="Strong equilibrium" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">strong equilibrium</a> result.</p><p style="margin:0.5em 0px;line-height:inherit">One interesting aspect in the above three scenarios: "assist" is only strategically favored in the case of a center squeeze scenario, when the A and C voters are helping B, whom they see as the the "lesser evil", beat the other extreme. If there isn't a center squeeze, voters need only use "support", "accept", or "reject". In fact, even in center squeeze, voters could get by without "assist", as long as enough A and C voters were willing to support B outright. So eliminating the "assist" option would not substantially change the strategic outcome; but it would reduce the expressive power available to the A and C voters in a center squeeze scenario. Thus majority score sacrifices some possible extra simplicity in return for this increased expressive power.</p><p style="margin:0.5em 0px;line-height:inherit">It is very rare to have a voting system which can deal with both chicken dilemma and center squeeze. The two situations are very similar, even when voted honestly, and yet the "correct" outcome is different. And under strategic voting in many voting systems, it is very easy for the two different scenarios to lead to identical ballots. Delegated voting systems such as <a href="http://wiki.electorama.com/wiki/SODA_voting" title="SODA voting" class="m_-182058607866077284gmail-mw-redirect" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">SODA voting</a> can deal with both; but without that kind of explicit participation in the voting process from the candidates, it is very hard to find a system which deals with both types of scenario better than majority score does.</p><h3 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;margin:0.3em 0px 0px;overflow:hidden;padding-top:0.5em;padding-bottom:0px;border-bottom:none;font-size:1.2em;line-height:1.6"><span class="m_-182058607866077284gmail-mw-headline" id="m_-182058607866077284gmail-Worst_case.3F">Worst case?</span><span class="m_-182058607866077284gmail-mw-editsection" style="font-size:small;font-weight:normal;margin-left:1em;vertical-align:baseline;line-height:1em;display:inline-block"><span class="m_-182058607866077284gmail-mw-editsection-bracket">[</span><a href="http://wiki.electorama.com/w/index.php?title=Majority_score_voting&action=edit&section=4" title="Edit section: Worst case?" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">edit</a><span class="m_-182058607866077284gmail-mw-editsection-bracket">]</span></span></h3><p style="margin:0.5em 0px;line-height:inherit">Majority score could still get a "wrong" answer in cases of a multi-way Chicken dilemma, such that none of the subfactions reached 25%. This scenario is avoidable if the subfactions compensate by "assisting" each other's candidates, but that cooperation is subject to slippery-slope chicken dynamics.</p><p style="margin:0.5em 0px;line-height:inherit">If this highly-specific scenario is the most-plausible worst case, it would seem that majority score is a pretty good system.</p><h2 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;font-weight:normal;margin:1em 0px 0.25em;overflow:hidden;padding:0px;border-bottom:1px solid rgb(170,170,170);font-family:"linux libertine",georgia,times,serif;line-height:1.3"><span class="m_-182058607866077284gmail-mw-headline" id="m_-182058607866077284gmail-As_the_first_round_of_a_two-round_system_.28.22majority_score_with_runoff.22.29">As the first round of a two-round system ("majority score with runoff")</span><span class="m_-182058607866077284gmail-mw-editsection" style="font-size:small;margin-left:1em;vertical-align:baseline;line-height:1em;display:inline-block;font-family:sans-serif"><span class="m_-182058607866077284gmail-mw-editsection-bracket">[</span><a href="http://wiki.electorama.com/w/index.php?title=Majority_score_voting&action=edit&section=5" title="Edit section: As the first round of a two-round system ("majority score with runoff")" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">edit</a><span class="m_-182058607866077284gmail-mw-editsection-bracket">]</span></span></h2><p style="margin:0.5em 0px;line-height:inherit">If this system is used as the first round of a two-round runoff, then you want to use it to elect at two finalists in the first round. Thus, run the system twice. The first time, replace "50%" in step 2 with "2/3".</p><p style="margin:0.5em 0px;line-height:inherit">Then, to find the second winner, if the first-time winner got 1/3 or more support, first downweight those ballots as if you'd eliminated enough of them to make up 1/3 of the electorate. Otherwise, discard all of the ballots which supported first-time winner. After downweighting or discarding, re-tally the points and run Majority Score again.</p><p style="margin:0.5em 0px;line-height:inherit">If all the candidates in the first round got a majority of 0's, then you can still find two finalists as explained above. But the voters have sent a message that none of the candidates are good, so one way to deal with the situation would be to have a rule to allow candidates to transfer their 2-votes to new candidates who were not running in the first round, and if those transfers would have made the new candidates finalists, then add them to the second round along with the two finalists who did best in the first round. In that case, since there would be more than two candidates in the second round, it would be important to use majority score for the second round too.</p><p style="margin:0.5em 0px;line-height:inherit">Note: this "proportional two-winner majority score" system for the first round is "matrix-summable", that is, summable with O(n²) information per ballot for n candidates. This contrasts with the base majority score method, which is summable with only O(n) information per ballot, and thus can be counted voting equipment designed for counting plurality elections.</p><h2 style="color:black;background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial;font-weight:normal;margin:1em 0px 0.25em;overflow:hidden;padding:0px;border-bottom:1px solid rgb(170,170,170);font-family:"linux libertine",georgia,times,serif;line-height:1.3"><span class="m_-182058607866077284gmail-mw-headline" id="m_-182058607866077284gmail-Relationship_to_NOTA">Relationship to NOTA</span><span class="m_-182058607866077284gmail-mw-editsection" style="font-size:small;margin-left:1em;vertical-align:baseline;line-height:1em;display:inline-block;font-family:sans-serif"><span class="m_-182058607866077284gmail-mw-editsection-bracket">[</span><a href="http://wiki.electorama.com/w/index.php?title=Majority_score_voting&action=edit&section=6" title="Edit section: Relationship to NOTA" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">edit</a><span class="m_-182058607866077284gmail-mw-editsection-bracket">]</span></span></h2><p style="margin:0.5em 0px;line-height:inherit">As discussed in the above section, if all the candidates in the first round got a majority "reject", then the voters have sent a message that none of the candidates are good, akin to a result of "<a href="http://wiki.electorama.com/wiki/None_of_the_above" title="None of the above" style="text-decoration:none;color:rgb(11,0,128);background-image:none;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;background-color:initial" target="_blank">none of the above</a>" (NOTA). Majority score still gives a winner, but it might be good to have a rule to limit the chance that such a winner would remain in office for multiple terms. This could either be a hard term limit, so that such a winner could only legally serve one term; or perhaps a softer rule that if they run for the same office again, the information of what percent of voters had rejected them should be next to their name on the ballot.</p></div></div></div></div></div><span class="HOEnZb"><font color="#888888">

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