{"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":"2021-02-05T23:54:54","modified_gmt":"2021-02-05T23:54:54","slug":"plurality-method","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/plurality-method\/","title":{"raw":"Plurality Method","rendered":"Plurality Method"},"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 preference ballots<\/li>\r\n \t<li>Evaluate the fairness\u00a0of an election using preference ballots<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Preference Schedules<\/h2>\r\nTo begin, we\u2019re going to want more information than a traditional ballot normally provides. A traditional ballot usually asks you to pick your favorite from a list of choices. This ballot fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.\r\n<div class=\"textbox\">\r\n<h3>Preference ballot<\/h3>\r\nA <strong>preference ballot<\/strong> is a ballot in which the voter ranks the choices in order of preference.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Data presentation<\/h3>\r\nIn the tables below, note the two variables present: city-order and number of votes. The first table lists the results of the vote. After tallying up the order preferences though, the voter name no longer matters. The second table lists how many votes each possible city-order received.\r\n\r\nWe can see that, given a list of three cities A, O, and H, there are 6 possible orderings that can be made.\r\n<p style=\"padding-left: 30px;\">The calculation to find the number of possible orderings from [latex]n[\/latex] given choices is [latex]n![\/latex]. The [latex]![\/latex] symbol is called <em>factorial<\/em>\u00a0. It is the product of the given integer [latex]n[\/latex] with each of the integers below it, up to and including [latex]1[\/latex]. In this case, [latex]3[\/latex] choices provide [latex]3 \\cdot 2 \\cdot 1 = 6[\/latex] choices. Here they are.<\/p>\r\n<p style=\"padding-left: 30px;\"><strong>AOH<\/strong>, OAH,<strong>OHA<\/strong>,<strong> AHO<\/strong>, <strong>HAO<\/strong>, HOA.<\/p>\r\nThere were two possible city-orders that received no votes: OAH and HOA. The four remaining orderings are listed in the second table under the number of votes each received.\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), Orlando (O), or Anaheim (A). Their votes are shown below:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>Bob<\/td>\r\n<td>Ann<\/td>\r\n<td>Marv<\/td>\r\n<td>Alice<\/td>\r\n<td>Eve<\/td>\r\n<td>Omar<\/td>\r\n<td>Lupe<\/td>\r\n<td>Dave<\/td>\r\n<td>Tish<\/td>\r\n<td>Jim<\/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<td>A<\/td>\r\n<td>O<\/td>\r\n<td>H<\/td>\r\n<td>O<\/td>\r\n<td>H<\/td>\r\n<td>A<\/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<td>H<\/td>\r\n<td>H<\/td>\r\n<td>A<\/td>\r\n<td>H<\/td>\r\n<td>A<\/td>\r\n<td>H<\/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<td>O<\/td>\r\n<td>A<\/td>\r\n<td>O<\/td>\r\n<td>A<\/td>\r\n<td>O<\/td>\r\n<td>O<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese individual ballots are typically combined into one <strong>preference schedule<\/strong>, which shows the number of voters in the top row that voted for each option:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>3<\/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\nNotice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: [latex]1+3+3+3=10[\/latex] total votes.\r\n\r\n<\/div>\r\nThe following video will give you a summary of what issues can arise from elections, as well as how a preference table is used in elections.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=WdtH_8lAqQo\r\n<h2>Plurality<\/h2>\r\nThe voting method we\u2019re most familiar with in the United States is the <strong>plurality method<\/strong>.\r\n<h3>Plurality Method<\/h3>\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 mistakenly called the majority method, or \u201cmajority rules\u201d, but it is not necessary for a choice to have gained a majority of votes to win. A majority is over 50%; it is possible for a winner to have a <strong>plurality<\/strong> without having a majority.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn our election from above, we had the preference table:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>3<\/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\nFor the plurality method, we only care about the first choice options. Totaling them up:\r\n\r\nAnaheim: [latex]1+3=4[\/latex] first-choice votes\r\n\r\nOrlando: 3 first-choice votes\r\n\r\nHawaii: 3 first-choice votes\r\n\r\nAnaheim is the winner using the plurality voting method.\r\n\r\nNotice that Anaheim won with 4 out of 10 votes, 40% of the votes, which is a plurality of the votes, but not a majority.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThree candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)[footnote]This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See <a href=\"http:\/\/www.co.pierce.wa.us\/xml\/abtus\/ourorg\/aud\/Elections\/RCV\/ranked\/exec\/summary.pdf\" target=\"_blank\" rel=\"noopener\">http:\/\/www.co.pierce.wa.us\/xml\/abtus\/ourorg\/aud\/Elections\/RCV\/ranked\/exec\/summary.pdf<\/a>[\/footnote]\u00a0The voting schedule is shown below. Which candidate wins under the plurality method?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>44<\/td>\r\n<td>14<\/td>\r\n<td>20<\/td>\r\n<td>70<\/td>\r\n<td>22<\/td>\r\n<td>80<\/td>\r\n<td>39<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>M<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td><\/td>\r\n<td>G<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td><\/td>\r\n<td>B<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote: In the third column and last column, those voters only recorded a first-place vote, so we don\u2019t know who their second and third choices would have been.\r\n\r\n[reveal-answer q=\"116686\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"116686\"]\r\n\r\nGoings received 78 1st-choice votes. McCarthy received 92. Bunney received 119. Bunney, who received approximately 41% of the 1st-choice votes is the plurality winner.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>What\u2019s Wrong with Plurality?<\/h2>\r\nThe election from the above example may seem totally clean, but there is a problem lurking that arises whenever there are three or more choices. Looking back at our preference table, how would our members vote if they only had two choices?\r\n\r\nAnaheim vs Orlando: 7 out of the 10 would prefer Anaheim over Orlando\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>3<\/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\nAnaheim vs Hawaii: 6 out of 10 would prefer Hawaii over Anaheim\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>3<\/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\nThis doesn\u2019t seem right, does it? Anaheim just won the election, yet 6 out of 10 voters, 60% of them, would have preferred Hawaii! That hardly seems fair. Marquis de Condorcet, a French philosopher, mathematician, and political scientist wrote about how this could happen in 1785, and for him we name our first <strong>fairness criterion<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>Fairness Criteria<\/h3>\r\nThe fairness criteria are statements that seem like they should be true in a fair election.\r\n\r\n<\/div>\r\n<h2>Condorcet Criterion<\/h2>\r\nIf there is a choice that is preferred in every one-to-one comparison with the other choices, that choice should be the winner. We call this winner the <strong>Condorcet Winner<\/strong>, or Condorcet Candidate.\r\n<div class=\"textbox examples\">\r\n<h3>Study strategy<\/h3>\r\nWork through the EXAMPLES and TRY IT problems below carefully, using pencil and paper, perhaps more than once.\r\n\r\nRemember, math must be performed to be understood well.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn the election, what choice is the Condorcet Winner?\r\n\r\nWe see above that Hawaii is preferred over Anaheim. Comparing Hawaii to Orlando, we can see 6 out of 10 would prefer Hawaii to Orlando.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>3<\/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\nSince Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider a city council election in a district that is historically 60% Democratic voters and 40% Republican voters. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. In this election there are three candidates: Don and Key, both Democrats, and Elle, a Republican. A preference schedule for the votes looks as follows:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>342<\/td>\r\n<td>214<\/td>\r\n<td>298<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>Elle<\/td>\r\n<td>Don<\/td>\r\n<td>Key<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>Don<\/td>\r\n<td>Key<\/td>\r\n<td>Don<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>Key<\/td>\r\n<td>Elle<\/td>\r\n<td>Elle<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can see a total of [latex]342+214+298=854[\/latex] voters participated in this election. Computing percentage of first place votes:\r\n\r\nDon: 214\/854 = 25.1%\r\n\r\nKey: 298\/854 = 34.9%\r\n\r\nElle: 342\/854 = 40.0%\r\n\r\nSo in this election, the Democratic voters split their vote over the two Democratic candidates, allowing the Republican candidate Elle to win under the plurality method with 40% of the vote.\r\n\r\nAnalyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons:\r\n\r\nElle vs Don: 342 prefer Elle; 512 prefer Don: Don is preferred\r\n\r\nElle vs Key: 342 prefer Elle; 512 prefer Key: Key is preferred\r\n\r\nDon vs Key: 556 prefer Don; 298 prefer Key: Don is preferred\r\n\r\nSo even though Don had the smallest number of first-place votes in the election, he is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.\r\n\r\n<\/div>\r\nIf you prefer to watch a video of the previous example being worked out, here it is.\r\n\r\nhttps:\/\/youtu.be\/x6DpoeaRVsw?list=PL1F887D3B8BF7C297\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider the election from the previous Try It. Is there a Condorcet winner in this election?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>44<\/td>\r\n<td>14<\/td>\r\n<td>20<\/td>\r\n<td>70<\/td>\r\n<td>22<\/td>\r\n<td>80<\/td>\r\n<td>39<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>M<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td><\/td>\r\n<td>G<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td><\/td>\r\n<td>B<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"736062\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"736062\"]\r\n\r\nBunney won the plurality with 119 votes to Goings's 78 votes and McCarthy's 92. The Condorcet method, though, examines head-to-head comparisons.\r\n\r\nBunney vs Goings:\u00a0102 prefer Bunney; 128 prefer Goings: Goings is preferred\r\n\r\nBunney vs McCarthy:\u00a094 prefer Bunney; 136 prefer McCarthy: McCarthy is preferred\r\n\r\nGoings vs McCarthy:\u00a058 prefer Goings; 172 prefer McCarthy: McCarthy is preferred.\r\n\r\nMcCarthy is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Insincere Voting<\/h2>\r\nSituations when there are more than one candidate that share somewhat similar points of view, can lead to <strong>insincere voting<\/strong>. Insincere voting is when a person casts a ballot counter to their actual preference for strategic purposes. In the case above, the democratic leadership might realize that Don and Key will split the vote, and encourage voters to vote for Key by officially endorsing him. Not wanting to see their party lose the election, as happened in the scenario above, Don\u2019s supporters might insincerely vote for Key, effectively voting against Elle.\r\n\r\nThe following video gives another mini lesson that covers the plurality method of voting as well as the idea of a Condorcet Winner.\r\n\r\nhttps:\/\/youtu.be\/r-VmxJQFMq8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the winner of an election using preference ballots<\/li>\n<li>Evaluate the fairness\u00a0of an election using preference ballots<\/li>\n<\/ul>\n<\/div>\n<h2>Preference Schedules<\/h2>\n<p>To begin, we\u2019re going to want more information than a traditional ballot normally provides. A traditional ballot usually asks you to pick your favorite from a list of choices. This ballot fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.<\/p>\n<div class=\"textbox\">\n<h3>Preference ballot<\/h3>\n<p>A <strong>preference ballot<\/strong> is a ballot in which the voter ranks the choices in order of preference.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Data presentation<\/h3>\n<p>In the tables below, note the two variables present: city-order and number of votes. The first table lists the results of the vote. After tallying up the order preferences though, the voter name no longer matters. The second table lists how many votes each possible city-order received.<\/p>\n<p>We can see that, given a list of three cities A, O, and H, there are 6 possible orderings that can be made.<\/p>\n<p style=\"padding-left: 30px;\">The calculation to find the number of possible orderings from [latex]n[\/latex] given choices is [latex]n![\/latex]. The [latex]![\/latex] symbol is called <em>factorial<\/em>\u00a0. It is the product of the given integer [latex]n[\/latex] with each of the integers below it, up to and including [latex]1[\/latex]. In this case, [latex]3[\/latex] choices provide [latex]3 \\cdot 2 \\cdot 1 = 6[\/latex] choices. Here they are.<\/p>\n<p style=\"padding-left: 30px;\"><strong>AOH<\/strong>, OAH,<strong>OHA<\/strong>,<strong> AHO<\/strong>, <strong>HAO<\/strong>, HOA.<\/p>\n<p>There were two possible city-orders that received no votes: OAH and HOA. The four remaining orderings are listed in the second table under the number of votes each received.<\/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), Orlando (O), or Anaheim (A). Their votes are shown below:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>Bob<\/td>\n<td>Ann<\/td>\n<td>Marv<\/td>\n<td>Alice<\/td>\n<td>Eve<\/td>\n<td>Omar<\/td>\n<td>Lupe<\/td>\n<td>Dave<\/td>\n<td>Tish<\/td>\n<td>Jim<\/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<td>A<\/td>\n<td>O<\/td>\n<td>H<\/td>\n<td>O<\/td>\n<td>H<\/td>\n<td>A<\/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<td>H<\/td>\n<td>H<\/td>\n<td>A<\/td>\n<td>H<\/td>\n<td>A<\/td>\n<td>H<\/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<td>O<\/td>\n<td>A<\/td>\n<td>O<\/td>\n<td>A<\/td>\n<td>O<\/td>\n<td>O<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These individual ballots are typically combined into one <strong>preference schedule<\/strong>, which shows the number of voters in the top row that voted for each option:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>3<\/td>\n<td>3<\/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>Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: [latex]1+3+3+3=10[\/latex] total votes.<\/p>\n<\/div>\n<p>The following video will give you a summary of what issues can arise from elections, as well as how a preference table is used 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<h2>Plurality<\/h2>\n<p>The voting method we\u2019re most familiar with in the United States is the <strong>plurality method<\/strong>.<\/p>\n<h3>Plurality Method<\/h3>\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 mistakenly called the majority method, or \u201cmajority rules\u201d, but it is not necessary for a choice to have gained a majority of votes to win. A majority is over 50%; it is possible for a winner to have a <strong>plurality<\/strong> without having a majority.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In our election from above, we had the preference table:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>3<\/td>\n<td>3<\/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>For the plurality method, we only care about the first choice options. Totaling them up:<\/p>\n<p>Anaheim: [latex]1+3=4[\/latex] first-choice votes<\/p>\n<p>Orlando: 3 first-choice votes<\/p>\n<p>Hawaii: 3 first-choice votes<\/p>\n<p>Anaheim is the winner using the plurality voting method.<\/p>\n<p>Notice that Anaheim won with 4 out of 10 votes, 40% of the votes, which is a plurality of the votes, but not a majority.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)<a class=\"footnote\" title=\"This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See http:\/\/www.co.pierce.wa.us\/xml\/abtus\/ourorg\/aud\/Elections\/RCV\/ranked\/exec\/summary.pdf\" id=\"return-footnote-1576-1\" href=\"#footnote-1576-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0The voting schedule is shown below. Which candidate wins under the plurality method?<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>44<\/td>\n<td>14<\/td>\n<td>20<\/td>\n<td>70<\/td>\n<td>22<\/td>\n<td>80<\/td>\n<td>39<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>M<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td><\/td>\n<td>G<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td><\/td>\n<td>B<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note: In the third column and last column, those voters only recorded a first-place vote, so we don\u2019t know who their second and third choices would have been.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q116686\">Solution<\/span><\/p>\n<div id=\"q116686\" class=\"hidden-answer\" style=\"display: none\">\n<p>Goings received 78 1st-choice votes. McCarthy received 92. Bunney received 119. Bunney, who received approximately 41% of the 1st-choice votes is the plurality winner.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>What\u2019s Wrong with Plurality?<\/h2>\n<p>The election from the above example may seem totally clean, but there is a problem lurking that arises whenever there are three or more choices. Looking back at our preference table, how would our members vote if they only had two choices?<\/p>\n<p>Anaheim vs Orlando: 7 out of the 10 would prefer Anaheim over Orlando<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>3<\/td>\n<td>3<\/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>Anaheim vs Hawaii: 6 out of 10 would prefer Hawaii over Anaheim<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>3<\/td>\n<td>3<\/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>This doesn\u2019t seem right, does it? Anaheim just won the election, yet 6 out of 10 voters, 60% of them, would have preferred Hawaii! That hardly seems fair. Marquis de Condorcet, a French philosopher, mathematician, and political scientist wrote about how this could happen in 1785, and for him we name our first <strong>fairness criterion<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>Fairness Criteria<\/h3>\n<p>The fairness criteria are statements that seem like they should be true in a fair election.<\/p>\n<\/div>\n<h2>Condorcet Criterion<\/h2>\n<p>If there is a choice that is preferred in every one-to-one comparison with the other choices, that choice should be the winner. We call this winner the <strong>Condorcet Winner<\/strong>, or Condorcet Candidate.<\/p>\n<div class=\"textbox examples\">\n<h3>Study strategy<\/h3>\n<p>Work through the EXAMPLES and TRY IT problems below carefully, using pencil and paper, perhaps more than once.<\/p>\n<p>Remember, math must be performed to be understood well.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In the election, what choice is the Condorcet Winner?<\/p>\n<p>We see above that Hawaii is preferred over Anaheim. Comparing Hawaii to Orlando, we can see 6 out of 10 would prefer Hawaii to Orlando.<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>3<\/td>\n<td>3<\/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>Since Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider a city council election in a district that is historically 60% Democratic voters and 40% Republican voters. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. In this election there are three candidates: Don and Key, both Democrats, and Elle, a Republican. A preference schedule for the votes looks as follows:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>342<\/td>\n<td>214<\/td>\n<td>298<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>Elle<\/td>\n<td>Don<\/td>\n<td>Key<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>Don<\/td>\n<td>Key<\/td>\n<td>Don<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>Key<\/td>\n<td>Elle<\/td>\n<td>Elle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can see a total of [latex]342+214+298=854[\/latex] voters participated in this election. Computing percentage of first place votes:<\/p>\n<p>Don: 214\/854 = 25.1%<\/p>\n<p>Key: 298\/854 = 34.9%<\/p>\n<p>Elle: 342\/854 = 40.0%<\/p>\n<p>So in this election, the Democratic voters split their vote over the two Democratic candidates, allowing the Republican candidate Elle to win under the plurality method with 40% of the vote.<\/p>\n<p>Analyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons:<\/p>\n<p>Elle vs Don: 342 prefer Elle; 512 prefer Don: Don is preferred<\/p>\n<p>Elle vs Key: 342 prefer Elle; 512 prefer Key: Key is preferred<\/p>\n<p>Don vs Key: 556 prefer Don; 298 prefer Key: Don is preferred<\/p>\n<p>So even though Don had the smallest number of first-place votes in the election, he is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.<\/p>\n<\/div>\n<p>If you prefer to watch a video of the previous example being worked out, here it is.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Condorcet winner and insincere voting with plurality method\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/x6DpoeaRVsw?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider the election from the previous Try It. Is there a Condorcet winner in this election?<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>44<\/td>\n<td>14<\/td>\n<td>20<\/td>\n<td>70<\/td>\n<td>22<\/td>\n<td>80<\/td>\n<td>39<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>M<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td><\/td>\n<td>G<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td><\/td>\n<td>B<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q736062\">Solution<\/span><\/p>\n<div id=\"q736062\" class=\"hidden-answer\" style=\"display: none\">\n<p>Bunney won the plurality with 119 votes to Goings&#8217;s 78 votes and McCarthy&#8217;s 92. The Condorcet method, though, examines head-to-head comparisons.<\/p>\n<p>Bunney vs Goings:\u00a0102 prefer Bunney; 128 prefer Goings: Goings is preferred<\/p>\n<p>Bunney vs McCarthy:\u00a094 prefer Bunney; 136 prefer McCarthy: McCarthy is preferred<\/p>\n<p>Goings vs McCarthy:\u00a058 prefer Goings; 172 prefer McCarthy: McCarthy is preferred.<\/p>\n<p>McCarthy is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Insincere Voting<\/h2>\n<p>Situations when there are more than one candidate that share somewhat similar points of view, can lead to <strong>insincere voting<\/strong>. Insincere voting is when a person casts a ballot counter to their actual preference for strategic purposes. In the case above, the democratic leadership might realize that Don and Key will split the vote, and encourage voters to vote for Key by officially endorsing him. Not wanting to see their party lose the election, as happened in the scenario above, Don\u2019s supporters might insincerely vote for Key, effectively voting against Elle.<\/p>\n<p>The following video gives another mini lesson that covers the plurality method of voting as well as the idea of a Condorcet Winner.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Voting Theory: Plurality Method and Condorcet Criterion\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/r-VmxJQFMq8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1576-1\">This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See <a href=\"http:\/\/www.co.pierce.wa.us\/xml\/abtus\/ourorg\/aud\/Elections\/RCV\/ranked\/exec\/summary.pdf\" target=\"_blank\" rel=\"noopener\">http:\/\/www.co.pierce.wa.us\/xml\/abtus\/ourorg\/aud\/Elections\/RCV\/ranked\/exec\/summary.pdf<\/a> <a href=\"#return-footnote-1576-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Voting Theory: Plurality Method and Condorcet Criterion\",\"author\":\"Sousa, James (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/r-VmxJQFMq8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Introduction to Voting Theory and Preference Tables\",\"author\":\"Sousa, James (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/6rhpq1ozmuQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Condorcet winner and insincere voting with plurality method\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/x6DpoeaRVsw?list=PL1F887D3B8BF7C297\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"ab1c42d4-ea9c-4e19-8ed9-dbd2a5e49ff6","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1576","chapter","type-chapter","status-web-only","hentry"],"part":1040,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1576","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1576\/revisions"}],"predecessor-version":[{"id":5332,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1576\/revisions\/5332"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/1040"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1576\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=1576"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1576"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=1576"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=1576"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}