{"id":3840,"date":"2023-05-31T15:59:44","date_gmt":"2023-05-31T15:59:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3840"},"modified":"2025-05-22T14:44:19","modified_gmt":"2025-05-22T14:44:19","slug":"voting-theory-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/voting-theory-learn-it-2\/","title":{"raw":"Voting Theory: Learn It 2","rendered":"Voting Theory: Learn It 2"},"content":{"raw":"<h2>Instant Runoff Voting<\/h2>\r\n<p>Instant Runoff Voting (IRV), also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting. In IRV, voting is done with preference ballots, and a preference schedule is generated. The choice with the <em>least<\/em> first-place votes is then eliminated from the election, and any votes for that candidate are redistributed to the voters\u2019 next choice. This continues until a choice has a majority (over [latex]50\\%[\/latex]).<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>instant runoff voting<\/h3>\r\n<p>In the instant runoff voting method, voters rank their preferred candidates or options in order of preference. <br \/>\r\n<br \/>\r\nIn each round of counting, the candidate with the fewest first-place votes is eliminated, and the votes for that candidate are redistributed to the voters' next choice. <br \/>\r\n<br \/>\r\nThis process continues until one candidate obtains an absolute majority of votes, usually exceeding [latex]50\\%[\/latex], and is declared the winner of the election.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>This is similar to the idea of holding runoff elections, but since every voter\u2019s order of preference is recorded on the ballot, the runoff can be computed without requiring a second costly election.<\/p>\r\n<p>This voting method is used in several political elections around the world, including election of members of the Australian House of Representatives, and was used for county positions in Pierce County, Washington until it was eliminated by voters in 2009. A version of IRV is used by the International Olympic Committee to select host nations.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider the preference schedule below, in which a company\u2019s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity.<\/p>\r\n<p>Initial votes<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>E<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>A<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>E<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>A<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>A<\/td>\r\n<td>E<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5th choice<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>If this was a plurality election, note that B would be the winner with [latex]9[\/latex] first-choice votes, compared to [latex]6[\/latex] for D, [latex]4[\/latex] for C, and [latex]1[\/latex] for E.<\/p>\r\n<p>There are total of [latex]3+4+4+6+2+1 = 20[\/latex] votes. A majority would be [latex]11[\/latex] votes. No one yet has a majority, so we proceed to elimination rounds.<\/p>\r\n<p><strong>Round 1<\/strong>: We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>E<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>E<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>E<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5th choice<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>We then shift everyone\u2019s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>E<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>D<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>E<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>E<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p><strong>Round 2<\/strong>: We make our second elimination. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone\u2019s options to fill the gaps.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>D<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>Notice that the first and fifth columns have the same preferences now, we can condense those down to one column.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>D<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Now B has [latex]9[\/latex] first-choice votes, C has [latex]4[\/latex] votes, and D has [latex]7[\/latex] votes. Still no majority, so we eliminate again.<\/p>\r\n<p><strong>Round 3<\/strong>: We make our third elimination. C has the fewest votes.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Condensing this down:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>D has now gained a majority, and is declared the winner under IRV.<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\">\r\n<p>The following video provides another\u00a0view of the example from above.<\/p>\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=12476201&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=kLbDqV9M3F0&amp;video_target=tpm-plugin-8mybvdil-kLbDqV9M3F0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Instant+Runoff+Method+IVR.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInstant Runoff Method IVR\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>What\u2019s Wrong with IRV?<\/h3>\r\n<section class=\"textbox example\">\r\n<p>Let\u2019s return to our City Council Election.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]342[\/latex]<\/td>\r\n<td>[latex]214[\/latex]<\/td>\r\n<td>[latex]298[\/latex]<\/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\n<p>&nbsp;<\/p>\r\n<p>In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The [latex]214[\/latex] people who voted for Don have their votes transferred to their second choice, Key.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]342[\/latex]<\/td>\r\n<td>[latex]512[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>Elle<\/td>\r\n<td>Key<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>Key<\/td>\r\n<td>Elle<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>So Key is the winner under the IRV method.<\/p>\r\n<p>We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don\u2019s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.<\/p>\r\n<\/section>\r\n<p>In the following video, we provide the example from above where we find that the IRV method violates the Condorcet Criterion in an election for a city council seat.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe src=\"https:\/\/www.youtube.com\/embed\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297\" width=\"1594\" height=\"897\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Instant+Runoff+Voting+2.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInstant Runoff Voting 2\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider the voting system below.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]37[\/latex]<\/td>\r\n<td>[latex]22[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]29[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>Adams<\/td>\r\n<td>Brown<\/td>\r\n<td>Brown<\/td>\r\n<td>Carter<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>Brown<\/td>\r\n<td>Carter<\/td>\r\n<td>Adams<\/td>\r\n<td>Adams<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>Carter<\/td>\r\n<td>Adams<\/td>\r\n<td>Carter<\/td>\r\n<td>Brown<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>In this election, Carter would be eliminated in the first round, and Adams would be the winner with [latex]66[\/latex] votes to [latex]34[\/latex] for Brown.<\/p>\r\n<p>Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to \u201cjump on the bandwagon,\u201d [latex]10[\/latex] of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]47[\/latex]<\/td>\r\n<td>[latex]22[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]29[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>Adams<\/td>\r\n<td>Brown<\/td>\r\n<td>Brown<\/td>\r\n<td>Carter<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>Brown<\/td>\r\n<td>Carter<\/td>\r\n<td>Adams<\/td>\r\n<td>Adams<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>Carter<\/td>\r\n<td>Adams<\/td>\r\n<td>Carter<\/td>\r\n<td>Brown<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with [latex]51[\/latex] votes to Adams\u2019 [latex]49[\/latex] votes! Even though the only vote changes made <em>favored<\/em> Adams, the change ended up costing Adams the election. This doesn\u2019t seem right, and introduces our second fairness criterion.<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>monotonicity criterion<\/h3>\r\n<p>If voters change their votes to increase the preference for a candidate, it should not harm that candidate\u2019s chances of winning.<\/p>\r\n<\/section>\r\n<p>This criterion is violated by this election. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; just that IRV has the potential to violate the criterion in certain elections.<\/p>\r\n<p>The last video shows the example from above where the monotonicity criterion is violated.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/youtu.be\/NH78zNXHKUs?list=PL1F887D3B8BF7C297[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/IRV+and+Monotonicity.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIRV and Monotonicity\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<h2>Instant Runoff Voting<\/h2>\n<p>Instant Runoff Voting (IRV), also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting. In IRV, voting is done with preference ballots, and a preference schedule is generated. The choice with the <em>least<\/em> first-place votes is then eliminated from the election, and any votes for that candidate are redistributed to the voters\u2019 next choice. This continues until a choice has a majority (over [latex]50\\%[\/latex]).<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>instant runoff voting<\/h3>\n<p>In the instant runoff voting method, voters rank their preferred candidates or options in order of preference. <\/p>\n<p>In each round of counting, the candidate with the fewest first-place votes is eliminated, and the votes for that candidate are redistributed to the voters&#8217; next choice. <\/p>\n<p>This process continues until one candidate obtains an absolute majority of votes, usually exceeding [latex]50\\%[\/latex], and is declared the winner of the election.<\/p>\n<\/div>\n<\/section>\n<p>This is similar to the idea of holding runoff elections, but since every voter\u2019s order of preference is recorded on the ballot, the runoff can be computed without requiring a second costly election.<\/p>\n<p>This voting method is used in several political elections around the world, including election of members of the Australian House of Representatives, and was used for county positions in Pierce County, Washington until it was eliminated by voters in 2009. A version of IRV is used by the International Olympic Committee to select host nations.<\/p>\n<section class=\"textbox example\">\n<p>Consider the preference schedule below, in which a company\u2019s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity.<\/p>\n<p>Initial votes<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>E<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>A<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>E<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>A<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>A<\/td>\n<td>E<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>5th choice<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>If this was a plurality election, note that B would be the winner with [latex]9[\/latex] first-choice votes, compared to [latex]6[\/latex] for D, [latex]4[\/latex] for C, and [latex]1[\/latex] for E.<\/p>\n<p>There are total of [latex]3+4+4+6+2+1 = 20[\/latex] votes. A majority would be [latex]11[\/latex] votes. No one yet has a majority, so we proceed to elimination rounds.<\/p>\n<p><strong>Round 1<\/strong>: We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>E<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>&nbsp;<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>E<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>&nbsp;<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>&nbsp;<\/td>\n<td>E<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>5th choice<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We then shift everyone\u2019s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>E<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>D<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>E<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>E<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Round 2<\/strong>: We make our second elimination. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone\u2019s options to fill the gaps.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>D<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the first and fifth columns have the same preferences now, we can condense those down to one column.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>D<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Now B has [latex]9[\/latex] first-choice votes, C has [latex]4[\/latex] votes, and D has [latex]7[\/latex] votes. Still no majority, so we eliminate again.<\/p>\n<p><strong>Round 3<\/strong>: We make our third elimination. C has the fewest votes.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>B<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Condensing this down:<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>D has now gained a majority, and is declared the winner under IRV.<\/p>\n<\/section>\n<section class=\"textbox watchIt\">\n<p>The following video provides another\u00a0view of the example from above.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=12476201&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=kLbDqV9M3F0&amp;video_target=tpm-plugin-8mybvdil-kLbDqV9M3F0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Instant+Runoff+Method+IVR.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInstant Runoff Method IVR\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>What\u2019s Wrong with IRV?<\/h3>\n<section class=\"textbox example\">\n<p>Let\u2019s return to our City Council Election.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]342[\/latex]<\/td>\n<td>[latex]214[\/latex]<\/td>\n<td>[latex]298[\/latex]<\/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>&nbsp;<\/p>\n<p>In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The [latex]214[\/latex] people who voted for Don have their votes transferred to their second choice, Key.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]342[\/latex]<\/td>\n<td>[latex]512[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>Elle<\/td>\n<td>Key<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>Key<\/td>\n<td>Elle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>So Key is the winner under the IRV method.<\/p>\n<p>We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don\u2019s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.<\/p>\n<\/section>\n<p>In the following video, we provide the example from above where we find that the IRV method violates the Condorcet Criterion in an election for a city council seat.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297\" width=\"1594\" height=\"897\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Instant+Runoff+Voting+2.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInstant Runoff Voting 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Consider the voting system below.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]37[\/latex]<\/td>\n<td>[latex]22[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]29[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>Adams<\/td>\n<td>Brown<\/td>\n<td>Brown<\/td>\n<td>Carter<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>Brown<\/td>\n<td>Carter<\/td>\n<td>Adams<\/td>\n<td>Adams<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>Carter<\/td>\n<td>Adams<\/td>\n<td>Carter<\/td>\n<td>Brown<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>In this election, Carter would be eliminated in the first round, and Adams would be the winner with [latex]66[\/latex] votes to [latex]34[\/latex] for Brown.<\/p>\n<p>Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to \u201cjump on the bandwagon,\u201d [latex]10[\/latex] of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]47[\/latex]<\/td>\n<td>[latex]22[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]29[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>Adams<\/td>\n<td>Brown<\/td>\n<td>Brown<\/td>\n<td>Carter<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>Brown<\/td>\n<td>Carter<\/td>\n<td>Adams<\/td>\n<td>Adams<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>Carter<\/td>\n<td>Adams<\/td>\n<td>Carter<\/td>\n<td>Brown<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with [latex]51[\/latex] votes to Adams\u2019 [latex]49[\/latex] votes! Even though the only vote changes made <em>favored<\/em> Adams, the change ended up costing Adams the election. This doesn\u2019t seem right, and introduces our second fairness criterion.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>monotonicity criterion<\/h3>\n<p>If voters change their votes to increase the preference for a candidate, it should not harm that candidate\u2019s chances of winning.<\/p>\n<\/section>\n<p>This criterion is violated by this election. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; just that IRV has the potential to violate the criterion in certain elections.<\/p>\n<p>The last video shows the example from above where the monotonicity criterion is violated.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"IRV and Monotonicity\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NH78zNXHKUs?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/IRV+and+Monotonicity.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIRV and Monotonicity\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":90,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Math in Society","author":"David Lippman","organization":"","url":"http:\/\/www.opentextbookstore.com\/mathinsociety\/","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3840"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":44,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3840\/revisions"}],"predecessor-version":[{"id":15498,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3840\/revisions\/15498"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/90"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3840\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3840"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3840"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3840"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}