{"id":1578,"date":"2017-02-16T18:37:08","date_gmt":"2017-02-16T18:37:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1578"},"modified":"2019-05-30T16:39:45","modified_gmt":"2019-05-30T16:39:45","slug":"instant-runoff-voting","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/instant-runoff-voting\/","title":{"raw":"Instant Runoff Voting","rendered":"Instant Runoff Voting"},"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 \t<li>Determine the winner of an election using the Instant Runoff method<\/li>\r\n \t<li>Evaluate the fairness\u00a0of an Instant Runoff election<\/li>\r\n \t<li>Determine the winner of an election using a Borda count<\/li>\r\n \t<li>Evaluate the fairness of an election determined using a Borda count<\/li>\r\n \t<li>Determine the winner of en election using Copeland's method<\/li>\r\n \t<li>Evaluate the fairness of an election determined by Copeland's method<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Instant Runoff Voting<\/h2>\r\nInstant 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 50%).\r\n\r\nThis 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.\r\n\r\nThis 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.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider 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.\r\n\r\nInitial votes\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>2<\/td>\r\n<td>1<\/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\nIf this was a plurality election, note that B would be the winner with 9 first-choice votes, compared to 6 for D, 4 for C, and 1 for E.\r\n\r\nThere are total of 3+4+4+6+2+1 = 20 votes. A majority would be 11 votes. No one yet has a majority, so we proceed to elimination rounds.\r\n\r\n<strong>Round 1<\/strong>: We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>2<\/td>\r\n<td>1<\/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><\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>E<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td><\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td><\/td>\r\n<td><\/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><\/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\nWe then shift everyone\u2019s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>2<\/td>\r\n<td>1<\/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<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.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>2<\/td>\r\n<td>1<\/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\nNotice that the first and fifth columns have the same preferences now, we can condense those down to one column.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>5<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>1<\/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\nNow B has 9 first-choice votes, C has 4 votes, and D has 7 votes. Still no majority, so we eliminate again.\r\n\r\n<strong>Round 3<\/strong>: We make our third elimination. C has the fewest votes.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>5<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>1<\/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\nCondensing this down:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>9<\/td>\r\n<td>11<\/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\nD has now gained a majority, and is declared the winner under IRV.\r\n\r\n<\/div>\r\nThe following video provides another\u00a0view of the example from above.\r\n\r\nhttps:\/\/youtu.be\/C-X-6Lo_xUQ?list=PL1F887D3B8BF7C297\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider again this election. Find the winner using IRV.\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<\/div>\r\nHere is an overview video that provides the definition of IRV, as well as an example of how to determine the winner of an election using IRV.\r\n\r\nhttps:\/\/youtu.be\/6axH6pcuyhQ\r\n<h3>What\u2019s Wrong with IRV?<\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLet\u2019s return to our City Council Election.\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\nIn this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The 214 people who voted for Don have their votes transferred to their second choice, Key.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>342<\/td>\r\n<td>512<\/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<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo Key is the winner under the IRV method.\r\n\r\nWe 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.\r\n\r\n<\/div>\r\nIn 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.\r\n\r\nhttps:\/\/youtu.be\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the voting system below.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>37<\/td>\r\n<td>22<\/td>\r\n<td>12<\/td>\r\n<td>29<\/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\nIn this election, Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 for Brown.\r\n\r\nNow 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 10 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.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>47<\/td>\r\n<td>22<\/td>\r\n<td>2<\/td>\r\n<td>29<\/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\nIn 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 51 votes to Adams\u2019 49 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:\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Monotonicity Criterion<\/h3>\r\nIf voters change their votes to increase the preference for a candidate, it should not harm that candidate\u2019s chances of winning.\r\n\r\n<\/div>\r\nThis 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.\r\n\r\nThe last video shows the example from above where the monotonicity criterion is violated.\r\n\r\nhttps:\/\/youtu.be\/NH78zNXHKUs?list=PL1F887D3B8BF7C297","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<li>Determine the winner of an election using the Instant Runoff method<\/li>\n<li>Evaluate the fairness\u00a0of an Instant Runoff election<\/li>\n<li>Determine the winner of an election using a Borda count<\/li>\n<li>Evaluate the fairness of an election determined using a Borda count<\/li>\n<li>Determine the winner of en election using Copeland&#8217;s method<\/li>\n<li>Evaluate the fairness of an election determined by Copeland&#8217;s method<\/li>\n<\/ul>\n<\/div>\n<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 50%).<\/p>\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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\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><\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>2<\/td>\n<td>1<\/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>If this was a plurality election, note that B would be the winner with 9 first-choice votes, compared to 6 for D, 4 for C, and 1 for E.<\/p>\n<p>There are total of 3+4+4+6+2+1 = 20 votes. A majority would be 11 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><\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>2<\/td>\n<td>1<\/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><\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>E<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td><\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td><\/td>\n<td><\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td><\/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><\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>2<\/td>\n<td>1<\/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><\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>2<\/td>\n<td>1<\/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><\/td>\n<td>5<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>1<\/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>Now B has 9 first-choice votes, C has 4 votes, and D has 7 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><\/td>\n<td>5<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>1<\/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>Condensing this down:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>9<\/td>\n<td>11<\/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>D has now gained a majority, and is declared the winner under IRV.<\/p>\n<\/div>\n<p>The following video provides another\u00a0view of the example from above.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Instant Runoff Voting\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/C-X-6Lo_xUQ?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider again this election. Find the winner using IRV.<\/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>\n<p>Here is an overview video that provides the definition of IRV, as well as an example of how to determine the winner of an election using IRV.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Voting Theory: Instant Runoff Voting\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6axH6pcuyhQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>What\u2019s Wrong with IRV?<\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Let\u2019s return to our City Council Election.<\/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>In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The 214 people who voted for Don have their votes transferred to their second choice, Key.<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>342<\/td>\n<td>512<\/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<tr>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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<\/div>\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<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Instant Runoff Voting 2\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the voting system below.<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>37<\/td>\n<td>22<\/td>\n<td>12<\/td>\n<td>29<\/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>In this election, Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 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 10 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><\/td>\n<td>47<\/td>\n<td>22<\/td>\n<td>2<\/td>\n<td>29<\/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>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 51 votes to Adams\u2019 49 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<\/div>\n<div class=\"textbox\">\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<\/div>\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<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"IRV and Monotonicity\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NH78zNXHKUs?list=PL1F887D3B8BF7C297\" 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-1578\">\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 Adaptaiton. <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: Instant Runoff Voting. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/6axH6pcuyhQ\">https:\/\/youtu.be\/6axH6pcuyhQ<\/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>Instant Runoff Voting . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/C-X-6Lo_xUQ?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/C-X-6Lo_xUQ?list=PL1F887D3B8BF7C297<\/a>. <strong>Project<\/strong>: Open Course Library. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Instant Runoff Voting 2 . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297<\/a>. <strong>Project<\/strong>: Open Course Library. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>IRV and Monotonicity . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/NH78zNXHKUs?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/NH78zNXHKUs?list=PL1F887D3B8BF7C297<\/a>. <strong>Project<\/strong>: Open Course Library. <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":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Voting Theory: Instant Runoff Voting\",\"author\":\"Sousa, James (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/6axH6pcuyhQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Instant Runoff Voting \",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/C-X-6Lo_xUQ?list=PL1F887D3B8BF7C297\",\"project\":\"Open Course Library\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Instant Runoff Voting 2 \",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/BCRaYCU28Ro?list=PL1F887D3B8BF7C297\",\"project\":\"Open Course 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