{"id":8847,"date":"2023-10-11T17:08:10","date_gmt":"2023-10-11T17:08:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=8847"},"modified":"2024-10-18T20:58:34","modified_gmt":"2024-10-18T20:58:34","slug":"voting-theory-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/voting-theory-learn-it-4\/","title":{"raw":"Voting Theory: Learn It 4","rendered":"Voting Theory: Learn It 4"},"content":{"raw":"

Copeland\u2019s Method<\/h2>\r\n

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>\r\n

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Copeland\u2019s method<\/h3>\r\n

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 [latex]1[\/latex] point. If there is a tie, each candidate is awarded [latex]\\frac{1}{2}[\/latex] point. After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.<\/p>\r\n<\/div>\r\n<\/section>\r\n

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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>\r\n<\/section>\r\n

Consider our vacation group example from the beginning of the module. A vacation club is trying to decide which destination to visit this year: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below. Determine the winner using Copeland\u2019s Method.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
1st choice<\/td>\r\nA<\/td>\r\nA<\/td>\r\nO<\/td>\r\nH<\/td>\r\n<\/tr>\r\n
2nd choice<\/td>\r\nO<\/td>\r\nH<\/td>\r\nH<\/td>\r\nA<\/td>\r\n<\/tr>\r\n
3rd choice<\/td>\r\nH<\/td>\r\nO<\/td>\r\nA<\/td>\r\nO<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[reveal-answer q=\"802469\"]Show Solution[\/reveal-answer] [hidden-answer a=\"802469\"]\r\n\r\n

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.<\/p>\r\n

For example, comparing Hawaii vs Orlando, we see that [latex]6[\/latex] 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 \u2014 we\u2019re imagining that Anaheim wasn\u2019t an option. If it helps, you can imagine removing Anaheim, as in the second table below.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
1st choice<\/td>\r\nA<\/td>\r\nA<\/td>\r\nO<\/td>\r\nH<\/td>\r\n<\/tr>\r\n
2nd choice<\/td>\r\nO<\/td>\r\nH<\/td>\r\nH<\/td>\r\nA<\/td>\r\n<\/tr>\r\n
3rd choice<\/td>\r\nH<\/td>\r\nO<\/td>\r\nA<\/td>\r\nO<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
1st choice<\/td>\r\n\u00a0<\/td>\r\n\u00a0<\/td>\r\nO<\/td>\r\nH<\/td>\r\n<\/tr>\r\n
2nd choice<\/td>\r\nO<\/td>\r\nH<\/td>\r\nH<\/td>\r\n\u00a0<\/td>\r\n<\/tr>\r\n
3rd choice<\/td>\r\nH<\/td>\r\nO<\/td>\r\n\u00a0<\/td>\r\nO<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

Based on this, in the comparison of Hawaii vs Orlando, Hawaii wins, and receives [latex]1[\/latex] point.<\/p>\r\n

Comparing Anaheim to Orlando, the [latex]1[\/latex] voter in the first column clearly prefers Anaheim, as do the [latex]3[\/latex] voters in the second column. The [latex]3[\/latex] voters in the third column clearly prefer Orlando.\u00a0 The [latex]3[\/latex] 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 [latex]3[\/latex] prefer Orlando over Anaheim. So, comparing Anaheim vs Orlando: [latex]7[\/latex] votes to [latex]3[\/latex] votes: Anaheim gets [latex]1[\/latex] point.<\/p>\r\n

All together,<\/p>\r\n

Hawaii vs Orlando: [latex]6[\/latex] votes to [latex]4[\/latex] votes: Hawaii gets [latex]1[\/latex] point<\/p>\r\n

Anaheim vs Orlando: [latex]7[\/latex] votes to [latex]3[\/latex] votes: Anaheim gets [latex]1[\/latex] point<\/p>\r\n

Hawaii vs Anaheim: [latex]6[\/latex] votes to [latex]4[\/latex] votes: Hawaii gets [latex]1[\/latex] point<\/p>\r\n

Hawaii is the winner under Copeland\u2019s Method, having earned the most points. Notice this process is consistent with our determination of a Condorcet Winner.<\/p>\r\n\r\nHere is the same example presented in a video.