{"id":1576,"date":"2017-02-16T18:30:52","date_gmt":"2017-02-16T18:30:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1576"},"modified":"2021-12-02T23:19:00","modified_gmt":"2021-12-02T23:19:00","slug":"plurality-method","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/coloradomesa-mathforliberalartscorequisite\/chapter\/plurality-method\/","title":{"raw":"Voting Methods: Majority Rule, Plurality, Single Run-off","rendered":"Voting Methods: Majority Rule, Plurality, Single Run-off"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine the winner of an election using majority rule.<\/li>\r\n \t<li>Determine the winner of an election using the plurality method.<\/li>\r\n \t<li>Construct a preference schedule.<\/li>\r\n \t<li>Determine the winner of an election using a single runoff.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Majority Rule<\/h2>\r\nIn the case that there are only two choices, the simplest voting method is\u00a0<strong>majority rule<\/strong>.\u00a0 Voters select one choice and the choice that receives more than 50% of the votes wins.\u00a0 Our U.S. Presidential Election is an example of majority rule, though it is not the majority of the population that is considered.\u00a0 Rather it is a simple majority of\u00a0<em>electoral votes<\/em> that determines the election outcome.\r\n<div class=\"textbox\">\r\n<h3>Majority Rule<\/h3>\r\n<strong>Majority rule<\/strong> is a voting method in which the winner must receive more than 50% of the votes.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA vacation club is trying to decide which destination to visit this year: Hawaii (H) or Anaheim (A). Their votes are shown below:\r\n<table style=\"width: 553px; height: 44px;\">\r\n<tbody>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 78px; height: 11px;\"><\/td>\r\n<td style=\"width: 38px; height: 11px;\">Bob<\/td>\r\n<td style=\"width: 39px; height: 11px;\">Ann<\/td>\r\n<td style=\"width: 46px; height: 11px;\">Marv<\/td>\r\n<td style=\"width: 46px; height: 11px;\">Alice<\/td>\r\n<td style=\"width: 39px; height: 11px;\">Eve<\/td>\r\n<td style=\"width: 52px; height: 11px;\">Omar<\/td>\r\n<td style=\"width: 46px; height: 11px;\">Lupe<\/td>\r\n<td style=\"width: 50px; height: 11px;\">Dave<\/td>\r\n<td style=\"width: 41px; height: 11px;\">Tish<\/td>\r\n<td style=\"width: 39px; height: 11px;\">Jim<\/td>\r\n<td style=\"width: 39px;\">Erika<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 78px; height: 11px;\">1st choice<\/td>\r\n<td style=\"width: 38px; height: 11px;\">A<\/td>\r\n<td style=\"width: 39px; height: 11px;\">A<\/td>\r\n<td style=\"width: 46px; height: 11px;\">H<\/td>\r\n<td style=\"width: 46px; height: 11px;\">H<\/td>\r\n<td style=\"width: 39px; height: 11px;\">A<\/td>\r\n<td style=\"width: 52px; height: 11px;\">H<\/td>\r\n<td style=\"width: 46px; height: 11px;\">H<\/td>\r\n<td style=\"width: 50px; height: 11px;\">H<\/td>\r\n<td style=\"width: 41px; height: 11px;\">H<\/td>\r\n<td style=\"width: 39px; height: 11px;\">A<\/td>\r\n<td style=\"width: 39px;\">A<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese individual ballots can be combined in a table\u00a0showing the number of voters in the top row that voted for each option:\r\n<table style=\"width: 327px; height: 33px;\">\r\n<tbody>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 364.656px; height: 11px;\">Number of votes<\/td>\r\n<td style=\"width: 137.656px; height: 11px;\">5<\/td>\r\n<td style=\"width: 109.656px; height: 11px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 364.656px; height: 11px;\">1st choice<\/td>\r\n<td style=\"width: 137.656px; height: 11px;\">A<\/td>\r\n<td style=\"width: 109.656px; height: 11px;\">H<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: [latex]5+6=11[\/latex] total votes.\u00a0 Since there are only two choices and Hawaii has won the majority (~55%) of the votes, then Hawaii is the winning destination.\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Plurality Method<\/h2>\r\nIn this method, the choice with the most first-preference votes is declared the winner. Ties are possible, and would have to be settled through some sort of run-off vote.\r\n\r\nThis method is sometimes assumed to be a \"Majority Rules\" method, but in fact, it is not necessary for a choice to have gained a majority of votes to win.\u00a0 A majority is over 50%; in a contest with more than two choices, it is possible for a winner to have a <strong>plurality<\/strong>\u00a0(the most first-preference votes) without having a majority.\r\n<div class=\"textbox\">\r\n<h3>Plurality Method<\/h3>\r\nThe\u00a0<strong>plurality method<\/strong> is a voting method in which the choice with the most votes wins.\u00a0 As noted above, the winning choice need not have captured a majority of votes in a content with more than two choices.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe following table illustrates the results of a four-way race for Psychology Club President.\u00a0 Notice that while Araceli received the most votes, no candidate received a majority (more than 50%) of the vote.\u00a0 Using the plurality method, Araceli would win the election.\r\n<table style=\"width: 486px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 237.656px;\"><\/td>\r\n<td style=\"width: 385.656px; text-align: center;\">Percentage of the Vote<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 237.656px;\">Liam<\/td>\r\n<td style=\"width: 385.656px; text-align: center;\">26%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 237.656px;\">Sophia<\/td>\r\n<td style=\"width: 385.656px; text-align: center;\">19%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 237.656px;\">Araceli<\/td>\r\n<td style=\"width: 385.656px; text-align: center;\">30%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 237.656px;\">Enrique<\/td>\r\n<td style=\"width: 385.656px; text-align: center;\">25%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHowever, there are some potential problems with the plurality method.\u00a0 Suppose, for example that all of the voters who chose Sophia and Enrique actually prefer Liam (over Araceli) as their second choice.\u00a0 Then, while Araceli would still have the most first place votes, Liam would actually be preferred over Araceli by\u00a0<strong>70%<\/strong> of the voters.\u00a0 One way to address this issue is to consider all of the voters' preferences.\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Preference Schedules<\/h3>\r\n<span style=\"font-size: 1rem; text-align: initial;\">A traditional ballot usually asks you to pick your favorite from a list of choices and fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.\u00a0 As we saw in the previous example, in the case of more than two candidates (or choices), this type of ballot can lead to an unfair outcome.\u00a0\u00a0A\u00a0<strong>preference ballot<\/strong>\u00a0is a ballot that records all of the voter's choices in order of preference.\u00a0\u00a0Tabulating a preference ballot requires a type of table called a\u00a0<strong>preference schedule<\/strong> that details the number of voters who chose each ranking order among the candidates.<\/span>\r\n\r\n&nbsp;\r\n\r\nThe following video gives an excellent summary of how the plurality method can lead to unfair results and how to construct and use a preference schedule in elections.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=WdtH_8lAqQo\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider again our vacation club.\u00a0 They have decided to consider a third possible destination and so now they are trying to decide between: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below:\r\n<table style=\"width: 646px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 102px;\"><\/td>\r\n<td style=\"width: 48px;\">Bob<\/td>\r\n<td style=\"width: 48px;\">Ann<\/td>\r\n<td style=\"width: 55px;\">Marv<\/td>\r\n<td style=\"width: 55px;\">Alice<\/td>\r\n<td style=\"width: 48px;\">Eve<\/td>\r\n<td style=\"width: 61px;\">Omar<\/td>\r\n<td style=\"width: 56px;\">Lupe<\/td>\r\n<td style=\"width: 59px;\">Dave<\/td>\r\n<td style=\"width: 50px;\">Tish<\/td>\r\n<td style=\"width: 45px;\">Jim<\/td>\r\n<td style=\"width: 19px;\">Erika<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 102px;\">1st choice<\/td>\r\n<td style=\"width: 48px;\">A<\/td>\r\n<td style=\"width: 48px;\">A<\/td>\r\n<td style=\"width: 55px;\">H<\/td>\r\n<td style=\"width: 55px;\">H<\/td>\r\n<td style=\"width: 48px;\">A<\/td>\r\n<td style=\"width: 61px;\">O<\/td>\r\n<td style=\"width: 56px;\">H<\/td>\r\n<td style=\"width: 59px;\">O<\/td>\r\n<td style=\"width: 50px;\">H<\/td>\r\n<td style=\"width: 45px;\">A<\/td>\r\n<td style=\"width: 19px;\">A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 102px;\">2nd choice<\/td>\r\n<td style=\"width: 48px;\">O<\/td>\r\n<td style=\"width: 48px;\">H<\/td>\r\n<td style=\"width: 55px;\">A<\/td>\r\n<td style=\"width: 55px;\">A<\/td>\r\n<td style=\"width: 48px;\">H<\/td>\r\n<td style=\"width: 61px;\">H<\/td>\r\n<td style=\"width: 56px;\">A<\/td>\r\n<td style=\"width: 59px;\">H<\/td>\r\n<td style=\"width: 50px;\">A<\/td>\r\n<td style=\"width: 45px;\">H<\/td>\r\n<td style=\"width: 19px;\">O<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 102px;\">3rd choice<\/td>\r\n<td style=\"width: 48px;\">H<\/td>\r\n<td style=\"width: 48px;\">O<\/td>\r\n<td style=\"width: 55px;\">O<\/td>\r\n<td style=\"width: 55px;\">O<\/td>\r\n<td style=\"width: 48px;\">O<\/td>\r\n<td style=\"width: 61px;\">A<\/td>\r\n<td style=\"width: 56px;\">O<\/td>\r\n<td style=\"width: 59px;\">A<\/td>\r\n<td style=\"width: 50px;\">O<\/td>\r\n<td style=\"width: 45px;\">O<\/td>\r\n<td style=\"width: 19px;\">H<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nThe individual ballots listed above are typically combined into one <strong>preference schedule\u00a0<\/strong>(table below), which shows the number of voters in the top row that voted for each option.\u00a0 You might notice that two of our six possible city-orders received no votes: OAH and HOA.\u00a0 The four remaining orderings are listed in the preference schedule below under the number of votes each received.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td>O<\/td>\r\n<td>H<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>O<\/td>\r\n<td>H<\/td>\r\n<td>H<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>H<\/td>\r\n<td>O<\/td>\r\n<td>A<\/td>\r\n<td>O<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nFor the plurality method, we only care about the first choice options. Totaling them up:\r\n\r\nAnaheim: [latex]2+3=5[\/latex] first-choice votes\r\n\r\nOrlando: 2 first-choice votes\r\n\r\nHawaii: 4 first-choice votes\r\n\r\nAnaheim is the winner using the plurality voting method.\r\n\r\nAgain, we notice that by totaling the vote counts across the top of the preference schedule we obtain the total number of votes cast: [latex]2+3+2+4=11[\/latex] total votes.\u00a0 Thus, we see that Anaheim won with 5 out of 11 votes, ~45% of the votes, <em>which is a plurality of the votes, but not a majority.<\/em>\r\n\r\n&nbsp;\r\n\r\nLooking again at our preference schedule, we observe that while Anaheim received the most first place votes (five), the other six people prefer Hawaii over Anaheim.\u00a0 It doesn't quite seem fair that Anaheim is the winning trip despite the fact that 6 out of 11 voters (~55%) would have preferred Hawaii!\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">\r\n<strong>Hawaii vs <\/strong><\/span><strong><span style=\"font-size: 1rem; text-align: initial;\">Anaheim: 6 to 5 preference for Hawaii over Anaheim<\/span><\/strong>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td><strong>2<\/strong><\/td>\r\n<td><strong>4<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td>O<\/td>\r\n<td><strong>H<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>O<\/td>\r\n<td>H<\/td>\r\n<td><strong>H<\/strong><\/td>\r\n<td><strong>A<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>H<\/td>\r\n<td>O<\/td>\r\n<td><strong>A<\/strong><\/td>\r\n<td>O<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nSince no option earned a majority (more than 50%) of first place votes, we can use the additional information included in our preference ballot to have a single run-off election.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Single Run-off Method<\/h3>\r\nIn a race with three or more choices in which no choice receives a majority, the\u00a0<strong>single run-off method<\/strong>\u00a0eliminates all but the top two first place vote-getters and then reevaluates the votes to determine the winner using majority rule.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn the vacation club example, we would eliminate Orlando, since Anaheim and Hawaii both received more than two votes.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td><del>O<\/del><\/td>\r\n<td>H<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td><del>O<\/del><\/td>\r\n<td>H<\/td>\r\n<td>H<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>H<\/td>\r\n<td><del>O<\/del><\/td>\r\n<td>A<\/td>\r\n<td><del>O<\/del><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nOur revised table would look like this:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td>H<\/td>\r\n<td>H<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>H<\/td>\r\n<td>H<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing this method, we see that Hawaii would win with a majority, 6 out of 11 votes (~55%).\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNote that the single run-off method is similar to simply holding a second run-off election, but since every voter\u2019s order of preference is recorded on the ballot, the runoff can be computed without requiring a separate (possibly costly) election.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the winner of an election using majority rule.<\/li>\n<li>Determine the winner of an election using the plurality method.<\/li>\n<li>Construct a preference schedule.<\/li>\n<li>Determine the winner of an election using a single runoff.<\/li>\n<\/ul>\n<\/div>\n<h2>Majority Rule<\/h2>\n<p>In the case that there are only two choices, the simplest voting method is\u00a0<strong>majority rule<\/strong>.\u00a0 Voters select one choice and the choice that receives more than 50% of the votes wins.\u00a0 Our U.S. Presidential Election is an example of majority rule, though it is not the majority of the population that is considered.\u00a0 Rather it is a simple majority of\u00a0<em>electoral votes<\/em> that determines the election outcome.<\/p>\n<div class=\"textbox\">\n<h3>Majority Rule<\/h3>\n<p><strong>Majority rule<\/strong> is a voting method in which the winner must receive more than 50% of the votes.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A vacation club is trying to decide which destination to visit this year: Hawaii (H) or Anaheim (A). Their votes are shown below:<\/p>\n<table style=\"width: 553px; height: 44px;\">\n<tbody>\n<tr style=\"height: 11px;\">\n<td style=\"width: 78px; height: 11px;\"><\/td>\n<td style=\"width: 38px; height: 11px;\">Bob<\/td>\n<td style=\"width: 39px; height: 11px;\">Ann<\/td>\n<td style=\"width: 46px; height: 11px;\">Marv<\/td>\n<td style=\"width: 46px; height: 11px;\">Alice<\/td>\n<td style=\"width: 39px; height: 11px;\">Eve<\/td>\n<td style=\"width: 52px; height: 11px;\">Omar<\/td>\n<td style=\"width: 46px; height: 11px;\">Lupe<\/td>\n<td style=\"width: 50px; height: 11px;\">Dave<\/td>\n<td style=\"width: 41px; height: 11px;\">Tish<\/td>\n<td style=\"width: 39px; height: 11px;\">Jim<\/td>\n<td style=\"width: 39px;\">Erika<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 78px; height: 11px;\">1st choice<\/td>\n<td style=\"width: 38px; height: 11px;\">A<\/td>\n<td style=\"width: 39px; height: 11px;\">A<\/td>\n<td style=\"width: 46px; height: 11px;\">H<\/td>\n<td style=\"width: 46px; height: 11px;\">H<\/td>\n<td style=\"width: 39px; height: 11px;\">A<\/td>\n<td style=\"width: 52px; height: 11px;\">H<\/td>\n<td style=\"width: 46px; height: 11px;\">H<\/td>\n<td style=\"width: 50px; height: 11px;\">H<\/td>\n<td style=\"width: 41px; height: 11px;\">H<\/td>\n<td style=\"width: 39px; height: 11px;\">A<\/td>\n<td style=\"width: 39px;\">A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These individual ballots can be combined in a table\u00a0showing the number of voters in the top row that voted for each option:<\/p>\n<table style=\"width: 327px; height: 33px;\">\n<tbody>\n<tr style=\"height: 11px;\">\n<td style=\"width: 364.656px; height: 11px;\">Number of votes<\/td>\n<td style=\"width: 137.656px; height: 11px;\">5<\/td>\n<td style=\"width: 109.656px; height: 11px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 364.656px; height: 11px;\">1st choice<\/td>\n<td style=\"width: 137.656px; height: 11px;\">A<\/td>\n<td style=\"width: 109.656px; height: 11px;\">H<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: [latex]5+6=11[\/latex] total votes.\u00a0 Since there are only two choices and Hawaii has won the majority (~55%) of the votes, then Hawaii is the winning destination.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Plurality Method<\/h2>\n<p>In this method, the choice with the most first-preference votes is declared the winner. Ties are possible, and would have to be settled through some sort of run-off vote.<\/p>\n<p>This method is sometimes assumed to be a &#8220;Majority Rules&#8221; method, but in fact, it is not necessary for a choice to have gained a majority of votes to win.\u00a0 A majority is over 50%; in a contest with more than two choices, it is possible for a winner to have a <strong>plurality<\/strong>\u00a0(the most first-preference votes) without having a majority.<\/p>\n<div class=\"textbox\">\n<h3>Plurality Method<\/h3>\n<p>The\u00a0<strong>plurality method<\/strong> is a voting method in which the choice with the most votes wins.\u00a0 As noted above, the winning choice need not have captured a majority of votes in a content with more than two choices.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The following table illustrates the results of a four-way race for Psychology Club President.\u00a0 Notice that while Araceli received the most votes, no candidate received a majority (more than 50%) of the vote.\u00a0 Using the plurality method, Araceli would win the election.<\/p>\n<table style=\"width: 486px;\">\n<tbody>\n<tr>\n<td style=\"width: 237.656px;\"><\/td>\n<td style=\"width: 385.656px; text-align: center;\">Percentage of the Vote<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 237.656px;\">Liam<\/td>\n<td style=\"width: 385.656px; text-align: center;\">26%<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 237.656px;\">Sophia<\/td>\n<td style=\"width: 385.656px; text-align: center;\">19%<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 237.656px;\">Araceli<\/td>\n<td style=\"width: 385.656px; text-align: center;\">30%<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 237.656px;\">Enrique<\/td>\n<td style=\"width: 385.656px; text-align: center;\">25%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>However, there are some potential problems with the plurality method.\u00a0 Suppose, for example that all of the voters who chose Sophia and Enrique actually prefer Liam (over Araceli) as their second choice.\u00a0 Then, while Araceli would still have the most first place votes, Liam would actually be preferred over Araceli by\u00a0<strong>70%<\/strong> of the voters.\u00a0 One way to address this issue is to consider all of the voters&#8217; preferences.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Preference Schedules<\/h3>\n<p><span style=\"font-size: 1rem; text-align: initial;\">A traditional ballot usually asks you to pick your favorite from a list of choices and fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.\u00a0 As we saw in the previous example, in the case of more than two candidates (or choices), this type of ballot can lead to an unfair outcome.\u00a0\u00a0A\u00a0<strong>preference ballot<\/strong>\u00a0is a ballot that records all of the voter&#8217;s choices in order of preference.\u00a0\u00a0Tabulating a preference ballot requires a type of table called a\u00a0<strong>preference schedule<\/strong> that details the number of voters who chose each ranking order among the candidates.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>The following video gives an excellent summary of how the plurality method can lead to unfair results and how to construct and use a preference schedule in elections.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Introduction to Voting Theory and Preference Tables\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/WdtH_8lAqQo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider again our vacation club.\u00a0 They have decided to consider a third possible destination and so now they are trying to decide between: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below:<\/p>\n<table style=\"width: 646px;\">\n<tbody>\n<tr>\n<td style=\"width: 102px;\"><\/td>\n<td style=\"width: 48px;\">Bob<\/td>\n<td style=\"width: 48px;\">Ann<\/td>\n<td style=\"width: 55px;\">Marv<\/td>\n<td style=\"width: 55px;\">Alice<\/td>\n<td style=\"width: 48px;\">Eve<\/td>\n<td style=\"width: 61px;\">Omar<\/td>\n<td style=\"width: 56px;\">Lupe<\/td>\n<td style=\"width: 59px;\">Dave<\/td>\n<td style=\"width: 50px;\">Tish<\/td>\n<td style=\"width: 45px;\">Jim<\/td>\n<td style=\"width: 19px;\">Erika<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 102px;\">1st choice<\/td>\n<td style=\"width: 48px;\">A<\/td>\n<td style=\"width: 48px;\">A<\/td>\n<td style=\"width: 55px;\">H<\/td>\n<td style=\"width: 55px;\">H<\/td>\n<td style=\"width: 48px;\">A<\/td>\n<td style=\"width: 61px;\">O<\/td>\n<td style=\"width: 56px;\">H<\/td>\n<td style=\"width: 59px;\">O<\/td>\n<td style=\"width: 50px;\">H<\/td>\n<td style=\"width: 45px;\">A<\/td>\n<td style=\"width: 19px;\">A<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 102px;\">2nd choice<\/td>\n<td style=\"width: 48px;\">O<\/td>\n<td style=\"width: 48px;\">H<\/td>\n<td style=\"width: 55px;\">A<\/td>\n<td style=\"width: 55px;\">A<\/td>\n<td style=\"width: 48px;\">H<\/td>\n<td style=\"width: 61px;\">H<\/td>\n<td style=\"width: 56px;\">A<\/td>\n<td style=\"width: 59px;\">H<\/td>\n<td style=\"width: 50px;\">A<\/td>\n<td style=\"width: 45px;\">H<\/td>\n<td style=\"width: 19px;\">O<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 102px;\">3rd choice<\/td>\n<td style=\"width: 48px;\">H<\/td>\n<td style=\"width: 48px;\">O<\/td>\n<td style=\"width: 55px;\">O<\/td>\n<td style=\"width: 55px;\">O<\/td>\n<td style=\"width: 48px;\">O<\/td>\n<td style=\"width: 61px;\">A<\/td>\n<td style=\"width: 56px;\">O<\/td>\n<td style=\"width: 59px;\">A<\/td>\n<td style=\"width: 50px;\">O<\/td>\n<td style=\"width: 45px;\">O<\/td>\n<td style=\"width: 19px;\">H<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>The individual ballots listed above are typically combined into one <strong>preference schedule\u00a0<\/strong>(table below), which shows the number of voters in the top row that voted for each option.\u00a0 You might notice that two of our six possible city-orders received no votes: OAH and HOA.\u00a0 The four remaining orderings are listed in the preference schedule below under the number of votes each received.<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>O<\/td>\n<td>H<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>O<\/td>\n<td>H<\/td>\n<td>H<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>H<\/td>\n<td>O<\/td>\n<td>A<\/td>\n<td>O<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>For the plurality method, we only care about the first choice options. Totaling them up:<\/p>\n<p>Anaheim: [latex]2+3=5[\/latex] first-choice votes<\/p>\n<p>Orlando: 2 first-choice votes<\/p>\n<p>Hawaii: 4 first-choice votes<\/p>\n<p>Anaheim is the winner using the plurality voting method.<\/p>\n<p>Again, we notice that by totaling the vote counts across the top of the preference schedule we obtain the total number of votes cast: [latex]2+3+2+4=11[\/latex] total votes.\u00a0 Thus, we see that Anaheim won with 5 out of 11 votes, ~45% of the votes, <em>which is a plurality of the votes, but not a majority.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>Looking again at our preference schedule, we observe that while Anaheim received the most first place votes (five), the other six people prefer Hawaii over Anaheim.\u00a0 It doesn&#8217;t quite seem fair that Anaheim is the winning trip despite the fact that 6 out of 11 voters (~55%) would have preferred Hawaii!<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><br \/>\n<strong>Hawaii vs <\/strong><\/span><strong><span style=\"font-size: 1rem; text-align: initial;\">Anaheim: 6 to 5 preference for Hawaii over Anaheim<\/span><\/strong><\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td><strong>2<\/strong><\/td>\n<td><strong>4<\/strong><\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>O<\/td>\n<td><strong>H<\/strong><\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>O<\/td>\n<td>H<\/td>\n<td><strong>H<\/strong><\/td>\n<td><strong>A<\/strong><\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>H<\/td>\n<td>O<\/td>\n<td><strong>A<\/strong><\/td>\n<td>O<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Since no option earned a majority (more than 50%) of first place votes, we can use the additional information included in our preference ballot to have a single run-off election.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Single Run-off Method<\/h3>\n<p>In a race with three or more choices in which no choice receives a majority, the\u00a0<strong>single run-off method<\/strong>\u00a0eliminates all but the top two first place vote-getters and then reevaluates the votes to determine the winner using majority rule.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In the vacation club example, we would eliminate Orlando, since Anaheim and Hawaii both received more than two votes.<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td><del>O<\/del><\/td>\n<td>H<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td><del>O<\/del><\/td>\n<td>H<\/td>\n<td>H<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>H<\/td>\n<td><del>O<\/del><\/td>\n<td>A<\/td>\n<td><del>O<\/del><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Our revised table would look like this:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>H<\/td>\n<td>H<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>H<\/td>\n<td>H<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using this method, we see that Hawaii would win with a majority, 6 out of 11 votes (~55%).<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Note that the single run-off method is similar to simply holding a second run-off election, but since every voter\u2019s order of preference is recorded on the ballot, the runoff can be computed without requiring a separate (possibly costly) election.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1576\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Voting Theory: Plurality Method and Condorcet Criterion. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/r-VmxJQFMq8\">https:\/\/youtu.be\/r-VmxJQFMq8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Introduction to Voting Theory and Preference Tables. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/6rhpq1ozmuQ\">https:\/\/youtu.be\/6rhpq1ozmuQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Condorcet winner and insincere voting with plurality method. <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/x6DpoeaRVsw?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/x6DpoeaRVsw?list=PL1F887D3B8BF7C297<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Voting Theory: Plurality Method and Condorcet Criterion\",\"author\":\"Sousa, James 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