{"id":1583,"date":"2017-02-16T18:44:59","date_gmt":"2017-02-16T18:44:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1583"},"modified":"2022-01-20T17:29:12","modified_gmt":"2022-01-20T17:29:12","slug":"copelands-method","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/copelands-method\/","title":{"raw":"Copeland's Method","rendered":"Copeland&#8217;s 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 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<div class=\"textbox examples\">\r\n<h3>Study Strategy<\/h3>\r\nWhen learning new vocabulary and processes it often takes more than a careful reading of the text to gain understanding. Remember to use flashcards for vocabulary, writing the answers out by hand before checking to see if you have them right. When learning new processes, writing them out by hand as you read through them will help you simultaneously memorize and gain insight into the process.\r\n\r\nPro-tip: Write out each of the examples in this section using paper and pencil, trying each of the steps as you go, until you feel you could explain it to another person.\r\n\r\n<\/div>\r\nSo far none of our voting methods have satisfied the Condorcet Criterion. The Copeland Method specifically attempts to satisfy the Condorcet Criterion by looking at pairwise (one-to-one) comparisons.\r\n<h3>Copeland\u2019s Method<\/h3>\r\nIn this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded \u00bd point. After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.\r\n\r\nVariations of Copeland\u2019s Method are used in many professional organizations, including election of the Board of Trustees for the Wikimedia Foundation that runs Wikipedia.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLet's refer back to the example given to determine a Condorcet winner. The following preference schedule came from an election whose candidates are Eric (E), Bridgette (B), Kelsie (K), and Daniel (D). Who would win using Copeland's method?\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\">29<\/td>\r\n<td style=\"width: 16.6667%;\">21<\/td>\r\n<td style=\"width: 16.6667%;\">18<\/td>\r\n<td style=\"width: 16.6667%;\">10<\/td>\r\n<td style=\"width: 16.6667%;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">1st choice<\/td>\r\n<td style=\"width: 16.6667%;\">D<\/td>\r\n<td style=\"width: 16.6667%;\">E<\/td>\r\n<td style=\"width: 16.6667%;\">B<\/td>\r\n<td style=\"width: 16.6667%;\">K<\/td>\r\n<td style=\"width: 16.6667%;\">K<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">2nd choice<\/td>\r\n<td style=\"width: 16.6667%;\">K<\/td>\r\n<td style=\"width: 16.6667%;\">K<\/td>\r\n<td style=\"width: 16.6667%;\">E<\/td>\r\n<td style=\"width: 16.6667%;\">B<\/td>\r\n<td style=\"width: 16.6667%;\">B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">3rd choice<\/td>\r\n<td style=\"width: 16.6667%;\">E<\/td>\r\n<td style=\"width: 16.6667%;\">B<\/td>\r\n<td style=\"width: 16.6667%;\">K<\/td>\r\n<td style=\"width: 16.6667%;\">E<\/td>\r\n<td style=\"width: 16.6667%;\">D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">4th choice<\/td>\r\n<td style=\"width: 16.6667%;\">B<\/td>\r\n<td style=\"width: 16.6667%;\">D<\/td>\r\n<td style=\"width: 16.6667%;\">D<\/td>\r\n<td style=\"width: 16.6667%;\">D<\/td>\r\n<td style=\"width: 16.6667%;\">E<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember that in using the Condorcet method, you want to examine head-to-head comparisons. You want to use this to determine which candidate is preferred and will receive the point.\r\n<ul>\r\n \t<li>Daniel vs Bridgette: 29 prefer Daniel; 50 (21+18+10+1) prefer Bridgette:\u00a0<em>Bridgette is preferred and receives 1 point.<\/em><\/li>\r\n \t<li>Daniel vs Kelsie: 29 prefer Daniel; 50 (21+18+10+1) prefer Kelsie:\u00a0<em>Kelsie is preferred and receives 1 point.<\/em><\/li>\r\n \t<li>Daniel vs Eric: 30 (29+1) prefer Daniel; 49 (21+18+10) prefer Eric:\u00a0<em>Eric is preferred and receives 1 point.<\/em><\/li>\r\n \t<li>Bridgette vs Kelsie: 18 prefer Bridgette; 61 (29+21+10+1) prefer Kelsie:\u00a0<em>Kelsie is preferred and receives 1 point.<\/em><\/li>\r\n \t<li>Bridgette vs Eric: 29 (18+10+1) prefer Bridgette; 50 (29+21) prefer Eric:\u00a0<em>Eric is preferred and receives 1 point.<\/em><\/li>\r\n \t<li>Kelsie vs Eric: 40 (29+10+1) prefer Kelsie; 39 (21+18) prefer Eric:\u00a0<em>Kelsie is preferred and receives 1 point.<\/em><\/li>\r\n<\/ul>\r\nUsing Copeland's method, Kelsie is the winner because she received 3 points.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider our vacation group example from the beginning of the chapter. Determine the winner using Copeland\u2019s Method.\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\n[reveal-answer q=\"802469\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"802469\"]\r\n\r\nWe need to look at each pair of choices, and see which choice would win in a one-to-one comparison. You may recall we did this earlier when determining the Condorcet Winner. For example, comparing Hawaii vs Orlando, we see that 6 voters, those shaded below in the first table below, would prefer Hawaii to Orlando. Note that Hawaii doesn\u2019t have to be the voter\u2019s first choice\u2014we\u2019re imagining that Anaheim wasn\u2019t an option. If it helps, you can imagine removing Anaheim, as in the second table below.\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\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><\/td>\r\n<td><\/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><\/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><\/td>\r\n<td>O<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBased on this, in the comparison of Hawaii vs Orlando, Hawaii wins, and receives 1 point.\r\n\r\nComparing Anaheim to Orlando, the 1 voter in the first column clearly prefers Anaheim, as do the 3 voters in the second column. The 3 voters in the third column clearly prefer Orlando.\u00a0 The 3 voters in the last column prefer Hawaii as their first choice, but if they had to choose between Anaheim and Orlando, they'd choose Anaheim, their second choice overall. So, altogether [latex]1+3+3=7[\/latex] voters prefer Anaheim over Orlando, and 3 prefer Orlando over Anaheim. So, comparing Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point.\r\n\r\nAll together,\r\n\r\nHawaii vs Orlando: 6 votes to 4 votes: Hawaii gets 1 point\r\n\r\nAnaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point\r\n\r\nHawaii vs Anaheim: 6 votes to 4 votes: Hawaii gets 1 point\r\n\r\nHawaii is the winner under Copeland\u2019s Method, having earned the most points.\r\n\r\nNotice this process is consistent with our determination of a Condorcet Winner.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is the same example presented in a video.\r\n\r\nhttps:\/\/youtu.be\/FftVPk7dqV0?list=PL1F887D3B8BF7C297\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the advertising group\u2019s vote we explored earlier. Determine the winner using Copeland\u2019s method.\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\n[reveal-answer q=\"944614\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944614\"]\r\n\r\nWith 5 candidates, there are 10 comparisons to make:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A vs B: 11 votes to 9 votes<\/td>\r\n<td>A gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs C: 3 votes to 17 votes<\/td>\r\n<td>C gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs D: 10 votes to 10 votes<\/td>\r\n<td>A gets \u00bd point, D gets \u00bd point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs E: 17 votes to 3 votes<\/td>\r\n<td>A gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs C: 10 votes to 10 votes<\/td>\r\n<td>B gets \u00bd point, C gets \u00bd point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs D: 9 votes to 11 votes<\/td>\r\n<td>D gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs E: 13 votes to 7 votes<\/td>\r\n<td>B gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C vs D: 9 votes to 11 votes<\/td>\r\n<td>D gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C vs E: 17 votes to 3 votes<\/td>\r\n<td>C gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D vs E: 17 votes to 3 votes<\/td>\r\n<td>D gets 1 point<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nTotaling these up:\r\n\r\nA gets 2\u00bd points\r\n\r\nB gets 1\u00bd points\r\n\r\nC gets 2\u00bd points\r\n\r\nD gets 3\u00bd points\r\n\r\nE gets 0 points\r\n\r\nUsing Copeland\u2019s Method, we declare D as the winner.\r\n\r\nNotice that in this case, D is not a Condorcet Winner. While Copeland\u2019s method will also select a Condorcet Candidate as the winner, the method still works in cases where there is no Condorcet Winner.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the same example from above being worked out in this video.\r\n\r\nhttps:\/\/youtu.be\/sWdmkee5m_Q?list=PL1F887D3B8BF7C297\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider again the election from earlier. Find the winner using Copeland\u2019s method. Since we have some incomplete preference ballots, we\u2019ll have to adjust. For example, when comparing M to B, we\u2019ll ignore the 20 votes in the third column which do not rank either candidate.\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=\"175839\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"175839\"]\r\n\r\nG vs M: 78 prefer G. 172 prefer M. M preferred and earns 1 point.\r\n\r\nG vs B: 148 prefer G. 141 prefer B. G preferred and earns 1 point.\r\n\r\nM vs B: 136 prefer M. 133 prefer B: M preferred and earns 1 point.\r\n\r\nM earns 2 points. G earns 1 point, and B earns 0 points. M wins under Copeland's method.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>What\u2019s Wrong with Copeland\u2019s Method?<\/h2>\r\nAs already noted, Copeland\u2019s Method does satisfy the Condorcet Criterion. It also satisfies the Majority Criterion and the Monotonicity Criterion. So is this the perfect method? Well, in a word, no.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA committee is trying to award a scholarship to one of four students, Anna (A), Brian (B), Carlos (C), and Dimitry (D). The votes are shown below:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>5<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>D<\/td>\r\n<td>A<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>A<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>A<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nMaking the comparisons:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A vs B: 10 votes to 10 votes<\/td>\r\n<td>A gets \u00bd point, B gets \u00bd point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs C: 14 votes to 6 votes:<\/td>\r\n<td>A gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs D: 5 votes to 15 votes:<\/td>\r\n<td>D gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs C: 4 votes to 16 votes:<\/td>\r\n<td>C gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs D: 15 votes to 5 votes:<\/td>\r\n<td>B gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C vs D: 11 votes to 9 votes:<\/td>\r\n<td>C gets 1 point<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTotaling:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A has 1 \u00bd points<\/td>\r\n<td>B has 1 \u00bd points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C has 2 points<\/td>\r\n<td>D has 1 point<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo Carlos is awarded the scholarship. However, the committee then discovers that Dimitry was not eligible for the scholarship (he failed his last math class). Even though this seems like it shouldn\u2019t affect the outcome, the committee decides to recount the vote, removing Dimitry from consideration. This reduces the preference schedule to:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>5<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>B<\/td>\r\n<td>B<\/td>\r\n<td>A<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A vs B: 10 votes to 10 votes<\/td>\r\n<td>A gets \u00bd point, B gets \u00bd point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs C: 14 votes to 6 votes<\/td>\r\n<td>A gets 1 point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs C: 4 votes to 16 votes<\/td>\r\n<td>C gets 1 point<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTotaling:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A has 1 \u00bd points<\/td>\r\n<td>B has \u00bd point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C has 1 point<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSuddenly Anna is the winner! This leads us to another fairness criterion.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>The Independence of Irrelevant Alternatives (IIA) Criterion<\/h3>\r\nIf a non-winning choice is removed from the ballot, it should not change the winner of the election.\r\n\r\nEquivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A.\r\n\r\n<\/div>\r\nIn the election from the last example, the IIA Criterion was violated.\r\n\r\nWatch this video to see the example from above worked out again,\r\n\r\nhttps:\/\/youtu.be\/463jDBNR-qY?list=PL1F887D3B8BF7C297\r\n\r\nThis anecdote illustrating the IIA issue is attributed to Sidney Morgenbesser:\r\n\r\nAfter finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says \"In that case I'll have the blueberry pie.\"\r\n\r\nAnother disadvantage of Copeland\u2019s Method is that it is fairly easy for the election to end in a tie. For this reason, Copeland\u2019s method is usually the first part of a more advanced method that uses more sophisticated methods for breaking ties and determining the winner when there is not a Condorcet Candidate.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the winner of an 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<div class=\"textbox examples\">\n<h3>Study Strategy<\/h3>\n<p>When learning new vocabulary and processes it often takes more than a careful reading of the text to gain understanding. Remember to use flashcards for vocabulary, writing the answers out by hand before checking to see if you have them right. When learning new processes, writing them out by hand as you read through them will help you simultaneously memorize and gain insight into the process.<\/p>\n<p>Pro-tip: Write out each of the examples in this section using paper and pencil, trying each of the steps as you go, until you feel you could explain it to another person.<\/p>\n<\/div>\n<p>So far none of our voting methods have satisfied the Condorcet Criterion. The Copeland Method specifically attempts to satisfy the Condorcet Criterion by looking at pairwise (one-to-one) comparisons.<\/p>\n<h3>Copeland\u2019s Method<\/h3>\n<p>In this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded \u00bd point. After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.<\/p>\n<p>Variations of Copeland\u2019s Method are used in many professional organizations, including election of the Board of Trustees for the Wikimedia Foundation that runs Wikipedia.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Let&#8217;s refer back to the example given to determine a Condorcet winner. The following preference schedule came from an election whose candidates are Eric (E), Bridgette (B), Kelsie (K), and Daniel (D). Who would win using Copeland&#8217;s method?<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\">29<\/td>\n<td style=\"width: 16.6667%;\">21<\/td>\n<td style=\"width: 16.6667%;\">18<\/td>\n<td style=\"width: 16.6667%;\">10<\/td>\n<td style=\"width: 16.6667%;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">1st choice<\/td>\n<td style=\"width: 16.6667%;\">D<\/td>\n<td style=\"width: 16.6667%;\">E<\/td>\n<td style=\"width: 16.6667%;\">B<\/td>\n<td style=\"width: 16.6667%;\">K<\/td>\n<td style=\"width: 16.6667%;\">K<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">2nd choice<\/td>\n<td style=\"width: 16.6667%;\">K<\/td>\n<td style=\"width: 16.6667%;\">K<\/td>\n<td style=\"width: 16.6667%;\">E<\/td>\n<td style=\"width: 16.6667%;\">B<\/td>\n<td style=\"width: 16.6667%;\">B<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">3rd choice<\/td>\n<td style=\"width: 16.6667%;\">E<\/td>\n<td style=\"width: 16.6667%;\">B<\/td>\n<td style=\"width: 16.6667%;\">K<\/td>\n<td style=\"width: 16.6667%;\">E<\/td>\n<td style=\"width: 16.6667%;\">D<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">4th choice<\/td>\n<td style=\"width: 16.6667%;\">B<\/td>\n<td style=\"width: 16.6667%;\">D<\/td>\n<td style=\"width: 16.6667%;\">D<\/td>\n<td style=\"width: 16.6667%;\">D<\/td>\n<td style=\"width: 16.6667%;\">E<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember that in using the Condorcet method, you want to examine head-to-head comparisons. You want to use this to determine which candidate is preferred and will receive the point.<\/p>\n<ul>\n<li>Daniel vs Bridgette: 29 prefer Daniel; 50 (21+18+10+1) prefer Bridgette:\u00a0<em>Bridgette is preferred and receives 1 point.<\/em><\/li>\n<li>Daniel vs Kelsie: 29 prefer Daniel; 50 (21+18+10+1) prefer Kelsie:\u00a0<em>Kelsie is preferred and receives 1 point.<\/em><\/li>\n<li>Daniel vs Eric: 30 (29+1) prefer Daniel; 49 (21+18+10) prefer Eric:\u00a0<em>Eric is preferred and receives 1 point.<\/em><\/li>\n<li>Bridgette vs Kelsie: 18 prefer Bridgette; 61 (29+21+10+1) prefer Kelsie:\u00a0<em>Kelsie is preferred and receives 1 point.<\/em><\/li>\n<li>Bridgette vs Eric: 29 (18+10+1) prefer Bridgette; 50 (29+21) prefer Eric:\u00a0<em>Eric is preferred and receives 1 point.<\/em><\/li>\n<li>Kelsie vs Eric: 40 (29+10+1) prefer Kelsie; 39 (21+18) prefer Eric:\u00a0<em>Kelsie is preferred and receives 1 point.<\/em><\/li>\n<\/ul>\n<p>Using Copeland&#8217;s method, Kelsie is the winner because she received 3 points.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider our vacation group example from the beginning of the chapter. Determine the winner using Copeland\u2019s Method.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q802469\">Show Solution<\/span><\/p>\n<div id=\"q802469\" class=\"hidden-answer\" style=\"display: none\">\n<p>We need to look at each pair of choices, and see which choice would win in a one-to-one comparison. You may recall we did this earlier when determining the Condorcet Winner. For example, comparing Hawaii vs Orlando, we see that 6 voters, those shaded below in the first table below, would prefer Hawaii to Orlando. Note that Hawaii doesn\u2019t have to be the voter\u2019s first choice\u2014we\u2019re imagining that Anaheim wasn\u2019t an option. If it helps, you can imagine removing Anaheim, as in the second table below.<\/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<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><\/td>\n<td><\/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><\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>H<\/td>\n<td>O<\/td>\n<td><\/td>\n<td>O<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Based on this, in the comparison of Hawaii vs Orlando, Hawaii wins, and receives 1 point.<\/p>\n<p>Comparing Anaheim to Orlando, the 1 voter in the first column clearly prefers Anaheim, as do the 3 voters in the second column. The 3 voters in the third column clearly prefer Orlando.\u00a0 The 3 voters in the last column prefer Hawaii as their first choice, but if they had to choose between Anaheim and Orlando, they&#8217;d choose Anaheim, their second choice overall. So, altogether [latex]1+3+3=7[\/latex] voters prefer Anaheim over Orlando, and 3 prefer Orlando over Anaheim. So, comparing Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point.<\/p>\n<p>All together,<\/p>\n<p>Hawaii vs Orlando: 6 votes to 4 votes: Hawaii gets 1 point<\/p>\n<p>Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point<\/p>\n<p>Hawaii vs Anaheim: 6 votes to 4 votes: Hawaii gets 1 point<\/p>\n<p>Hawaii is the winner under Copeland\u2019s Method, having earned the most points.<\/p>\n<p>Notice this process is consistent with our determination of a Condorcet Winner.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here is the same example presented in a video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Copeland&#39;s method \/ Pairwise comparison 1\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/FftVPk7dqV0?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the advertising group\u2019s vote we explored earlier. Determine the winner using Copeland\u2019s method.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944614\">Show Solution<\/span><\/p>\n<div id=\"q944614\" class=\"hidden-answer\" style=\"display: none\">\n<p>With 5 candidates, there are 10 comparisons to make:<\/p>\n<table>\n<tbody>\n<tr>\n<td>A vs B: 11 votes to 9 votes<\/td>\n<td>A gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>A vs C: 3 votes to 17 votes<\/td>\n<td>C gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>A vs D: 10 votes to 10 votes<\/td>\n<td>A gets \u00bd point, D gets \u00bd point<\/td>\n<\/tr>\n<tr>\n<td>A vs E: 17 votes to 3 votes<\/td>\n<td>A gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>B vs C: 10 votes to 10 votes<\/td>\n<td>B gets \u00bd point, C gets \u00bd point<\/td>\n<\/tr>\n<tr>\n<td>B vs D: 9 votes to 11 votes<\/td>\n<td>D gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>B vs E: 13 votes to 7 votes<\/td>\n<td>B gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>C vs D: 9 votes to 11 votes<\/td>\n<td>D gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>C vs E: 17 votes to 3 votes<\/td>\n<td>C gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>D vs E: 17 votes to 3 votes<\/td>\n<td>D gets 1 point<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Totaling these up:<\/p>\n<p>A gets 2\u00bd points<\/p>\n<p>B gets 1\u00bd points<\/p>\n<p>C gets 2\u00bd points<\/p>\n<p>D gets 3\u00bd points<\/p>\n<p>E gets 0 points<\/p>\n<p>Using Copeland\u2019s Method, we declare D as the winner.<\/p>\n<p>Notice that in this case, D is not a Condorcet Winner. While Copeland\u2019s method will also select a Condorcet Candidate as the winner, the method still works in cases where there is no Condorcet Winner.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the same example from above being worked out in this video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Copeland&#39;s method \/ Pairwise comparsion 2\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/sWdmkee5m_Q?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider again the election from earlier. Find the winner using Copeland\u2019s method. Since we have some incomplete preference ballots, we\u2019ll have to adjust. For example, when comparing M to B, we\u2019ll ignore the 20 votes in the third column which do not rank either candidate.<\/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=\"q175839\">Solution<\/span><\/p>\n<div id=\"q175839\" class=\"hidden-answer\" style=\"display: none\">\n<p>G vs M: 78 prefer G. 172 prefer M. M preferred and earns 1 point.<\/p>\n<p>G vs B: 148 prefer G. 141 prefer B. G preferred and earns 1 point.<\/p>\n<p>M vs B: 136 prefer M. 133 prefer B: M preferred and earns 1 point.<\/p>\n<p>M earns 2 points. G earns 1 point, and B earns 0 points. M wins under Copeland&#8217;s method.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>What\u2019s Wrong with Copeland\u2019s Method?<\/h2>\n<p>As already noted, Copeland\u2019s Method does satisfy the Condorcet Criterion. It also satisfies the Majority Criterion and the Monotonicity Criterion. So is this the perfect method? Well, in a word, no.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A committee is trying to award a scholarship to one of four students, Anna (A), Brian (B), Carlos (C), and Dimitry (D). The votes are shown below:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>5<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>D<\/td>\n<td>A<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>A<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>A<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Making the comparisons:<\/p>\n<table>\n<tbody>\n<tr>\n<td>A vs B: 10 votes to 10 votes<\/td>\n<td>A gets \u00bd point, B gets \u00bd point<\/td>\n<\/tr>\n<tr>\n<td>A vs C: 14 votes to 6 votes:<\/td>\n<td>A gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>A vs D: 5 votes to 15 votes:<\/td>\n<td>D gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>B vs C: 4 votes to 16 votes:<\/td>\n<td>C gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>B vs D: 15 votes to 5 votes:<\/td>\n<td>B gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>C vs D: 11 votes to 9 votes:<\/td>\n<td>C gets 1 point<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Totaling:<\/p>\n<table>\n<tbody>\n<tr>\n<td>A has 1 \u00bd points<\/td>\n<td>B has 1 \u00bd points<\/td>\n<\/tr>\n<tr>\n<td>C has 2 points<\/td>\n<td>D has 1 point<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So Carlos is awarded the scholarship. However, the committee then discovers that Dimitry was not eligible for the scholarship (he failed his last math class). Even though this seems like it shouldn\u2019t affect the outcome, the committee decides to recount the vote, removing Dimitry from consideration. This reduces the preference schedule to:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>5<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>B<\/td>\n<td>B<\/td>\n<td>A<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td>A vs B: 10 votes to 10 votes<\/td>\n<td>A gets \u00bd point, B gets \u00bd point<\/td>\n<\/tr>\n<tr>\n<td>A vs C: 14 votes to 6 votes<\/td>\n<td>A gets 1 point<\/td>\n<\/tr>\n<tr>\n<td>B vs C: 4 votes to 16 votes<\/td>\n<td>C gets 1 point<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Totaling:<\/p>\n<table>\n<tbody>\n<tr>\n<td>A has 1 \u00bd points<\/td>\n<td>B has \u00bd point<\/td>\n<\/tr>\n<tr>\n<td>C has 1 point<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Suddenly Anna is the winner! This leads us to another fairness criterion.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>The Independence of Irrelevant Alternatives (IIA) Criterion<\/h3>\n<p>If a non-winning choice is removed from the ballot, it should not change the winner of the election.<\/p>\n<p>Equivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A.<\/p>\n<\/div>\n<p>In the election from the last example, the IIA Criterion was violated.<\/p>\n<p>Watch this video to see the example from above worked out again,<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Copelands and the IIA criterion\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/463jDBNR-qY?list=PL1F887D3B8BF7C297\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This anecdote illustrating the IIA issue is attributed to Sidney Morgenbesser:<\/p>\n<p>After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says &#8220;In that case I&#8217;ll have the blueberry pie.&#8221;<\/p>\n<p>Another disadvantage of Copeland\u2019s Method is that it is fairly easy for the election to end in a tie. For this reason, Copeland\u2019s method is usually the first part of a more advanced method that uses more sophisticated methods for breaking ties and determining the winner when there is not a Condorcet Candidate.<\/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-1583\">\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>Copeland&#039;s method \/ Pairwise comparison 1 . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/FftVPk7dqV0?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/FftVPk7dqV0?list=PL1F887D3B8BF7C297<\/a>. <strong>Project<\/strong>: Open Course Library. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Copeland&#039;s method \/ Pairwise comparsion 2 . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/sWdmkee5m_Q?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/sWdmkee5m_Q?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>Copelands and the IIA criterion . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/463jDBNR-qY?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/463jDBNR-qY?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":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Copeland\\'s method \/ Pairwise comparison 1 \",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/FftVPk7dqV0?list=PL1F887D3B8BF7C297\",\"project\":\"Open Course Library\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Copeland\\'s method \/ Pairwise comparsion 2 \",\"author\":\"Lippman, 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