{"id":8840,"date":"2023-10-11T15:04:06","date_gmt":"2023-10-11T15:04:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8840"},"modified":"2024-10-18T20:58:33","modified_gmt":"2024-10-18T20:58:33","slug":"voting-theory-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/voting-theory-learn-it-3\/","title":{"raw":"Voting Theory: Learn It 3","rendered":"Voting Theory: Learn It 3"},"content":{"raw":"<h2>Borda Count<\/h2>\r\n<p><strong>Borda count<\/strong> is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Borda count<\/h3>\r\n<p>In this method, points are assigned to candidates based on their ranking; [latex]1[\/latex] point for last choice, [latex]2[\/latex] points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>A group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia. Their approximate locations on a map are shown below.<\/p>\r\n<center><img class=\"aligncenter wp-image-1581\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/16183951\/Puget-Sound.png\" alt=\"A map of the Puget Sound. Seattle, Tacoma, Puyallup, and Olympia are marked.\" width=\"258\" height=\"249\" \/><\/center>\r\n<p>The votes for where to hold the conference were:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]51[\/latex]<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>Seattle<\/td>\r\n<td>Tacoma<\/td>\r\n<td>Puyallup<\/td>\r\n<td>Olympia<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>Tacoma<\/td>\r\n<td>Puyallup<\/td>\r\n<td>Tacoma<\/td>\r\n<td>Tacoma<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>Olympia<\/td>\r\n<td>Olympia<\/td>\r\n<td>Olympia<\/td>\r\n<td>Puyallup<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice<\/td>\r\n<td>Puyallup<\/td>\r\n<td>Seattle<\/td>\r\n<td>Seattle<\/td>\r\n<td>Seattle<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Use the Borda count method to determine the winning town for the conference. [reveal-answer q=\"75557\"]Show Solution[\/reveal-answer] [hidden-answer a=\"75557\"] In each of the [latex]51[\/latex] ballots ranking Seattle first, Puyallup will be given [latex]1[\/latex] point, Olympia [latex]2[\/latex] points, Tacoma [latex]3[\/latex] points, and Seattle [latex]4[\/latex] points. Multiplying the points per vote times the number of votes allows us to calculate points awarded<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]51[\/latex]<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice [latex]4[\/latex] points<\/td>\r\n<td>Seattle [latex]4\\cdot51=204[\/latex]<\/td>\r\n<td>Tacoma [latex]4\\cdot25=100[\/latex]<\/td>\r\n<td>Puyallup [latex]4\\cdot10=40[\/latex]<\/td>\r\n<td>Olympia [latex]4\\cdot14=56[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice [latex]3[\/latex] points<\/td>\r\n<td>Tacoma [latex]3\\cdot51=153[\/latex]<\/td>\r\n<td>Puyallup [latex]3\\cdot25=75[\/latex]<\/td>\r\n<td>Tacoma [latex]3\\cdot10=30[\/latex]<\/td>\r\n<td>Tacoma [latex]3\\cdot14=42[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice [latex]2[\/latex] points<\/td>\r\n<td>Olympia [latex]2\\cdot51=102[\/latex]<\/td>\r\n<td>Olympia [latex]2\\cdot25=50[\/latex]<\/td>\r\n<td>Olympia [latex]2\\cdot10=20[\/latex]<\/td>\r\n<td>Puyallup [latex]2\\cdot14=28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice [latex]1[\/latex] point<\/td>\r\n<td>Puyallup [latex]1\\cdot51=51[\/latex]<\/td>\r\n<td>Seattle [latex]1\\cdot25=25[\/latex]<\/td>\r\n<td>Seattle [latex]1\\cdot10=10[\/latex]<\/td>\r\n<td>Seattle [latex]1\\cdot14=14[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Adding up the points:<\/p>\r\n<ul>\r\n\t<li>Seattle: [latex]204+25+10+14=253[\/latex] points<\/li>\r\n\t<li>Tacoma: [latex]153+100+30+42=325[\/latex] points<\/li>\r\n\t<li>Puyallup: [latex]51+75+40+28=194[\/latex] points<\/li>\r\n\t<li>Olympia: [latex]102+50+20+56=228[\/latex] points<\/li>\r\n<\/ul>\r\n<p>Under the Borda Count method, Tacoma is the winner of this vote. For more information on this problem, watch the following video.<\/p>\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vfujywLdW_s?si=UBQ_bMdAELaWcbNC\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Borda+Count.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBorda Count\u201d here (opens in new window).<\/a><\/p>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]12858[\/ohm2_question]<\/section>\r\n<h3>What\u2019s Wrong with Borda Count?<\/h3>\r\n<p>You might have already noticed one potential flaw of the Borda count from the previous example. In that example, Seattle had a majority of first-choice votes, yet lost the election! This seems odd, and prompts our next fairness criterion:<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>majority criterion<\/h3>\r\n<p>If a choice has a majority of first-place votes, that choice should be the winner.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>The election from the previous example using the Borda count violates the majority criterion. Notice also that this automatically means that the Condorcet Criterion will also be violated, as Seattle would have been preferred by [latex]51\\%[\/latex] of voters in any head-to-head comparison. Borda count is sometimes described as a consensus-based voting system, since it can sometimes choose a more broadly acceptable option over the one with majority support. In the example above, Tacoma is probably the best compromise location. This is a different approach than plurality and instant runoff voting that focus on first-choice votes; Borda count considers every voter\u2019s entire ranking to determine the outcome. Because of this consensus behavior, Borda count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy.<\/p>","rendered":"<h2>Borda Count<\/h2>\n<p><strong>Borda count<\/strong> is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Borda count<\/h3>\n<p>In this method, points are assigned to candidates based on their ranking; [latex]1[\/latex] point for last choice, [latex]2[\/latex] points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>A group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia. Their approximate locations on a map are shown below.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1581\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/16183951\/Puget-Sound.png\" alt=\"A map of the Puget Sound. Seattle, Tacoma, Puyallup, and Olympia are marked.\" width=\"258\" height=\"249\" \/><\/div>\n<p>The votes for where to hold the conference were:<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]51[\/latex]<\/td>\n<td>[latex]25[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>Seattle<\/td>\n<td>Tacoma<\/td>\n<td>Puyallup<\/td>\n<td>Olympia<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>Tacoma<\/td>\n<td>Puyallup<\/td>\n<td>Tacoma<\/td>\n<td>Tacoma<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>Olympia<\/td>\n<td>Olympia<\/td>\n<td>Olympia<\/td>\n<td>Puyallup<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>Puyallup<\/td>\n<td>Seattle<\/td>\n<td>Seattle<\/td>\n<td>Seattle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Use the Borda count method to determine the winning town for the conference. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q75557\">Show Solution<\/button> <\/p>\n<div id=\"q75557\" class=\"hidden-answer\" style=\"display: none\"> In each of the [latex]51[\/latex] ballots ranking Seattle first, Puyallup will be given [latex]1[\/latex] point, Olympia [latex]2[\/latex] points, Tacoma [latex]3[\/latex] points, and Seattle [latex]4[\/latex] points. Multiplying the points per vote times the number of votes allows us to calculate points awarded<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]51[\/latex]<\/td>\n<td>[latex]25[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice [latex]4[\/latex] points<\/td>\n<td>Seattle [latex]4\\cdot51=204[\/latex]<\/td>\n<td>Tacoma [latex]4\\cdot25=100[\/latex]<\/td>\n<td>Puyallup [latex]4\\cdot10=40[\/latex]<\/td>\n<td>Olympia [latex]4\\cdot14=56[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2nd choice [latex]3[\/latex] points<\/td>\n<td>Tacoma [latex]3\\cdot51=153[\/latex]<\/td>\n<td>Puyallup [latex]3\\cdot25=75[\/latex]<\/td>\n<td>Tacoma [latex]3\\cdot10=30[\/latex]<\/td>\n<td>Tacoma [latex]3\\cdot14=42[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3rd choice [latex]2[\/latex] points<\/td>\n<td>Olympia [latex]2\\cdot51=102[\/latex]<\/td>\n<td>Olympia [latex]2\\cdot25=50[\/latex]<\/td>\n<td>Olympia [latex]2\\cdot10=20[\/latex]<\/td>\n<td>Puyallup [latex]2\\cdot14=28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>4th choice [latex]1[\/latex] point<\/td>\n<td>Puyallup [latex]1\\cdot51=51[\/latex]<\/td>\n<td>Seattle [latex]1\\cdot25=25[\/latex]<\/td>\n<td>Seattle [latex]1\\cdot10=10[\/latex]<\/td>\n<td>Seattle [latex]1\\cdot14=14[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Adding up the points:<\/p>\n<ul>\n<li>Seattle: [latex]204+25+10+14=253[\/latex] points<\/li>\n<li>Tacoma: [latex]153+100+30+42=325[\/latex] points<\/li>\n<li>Puyallup: [latex]51+75+40+28=194[\/latex] points<\/li>\n<li>Olympia: [latex]102+50+20+56=228[\/latex] points<\/li>\n<\/ul>\n<p>Under the Borda Count method, Tacoma is the winner of this vote. For more information on this problem, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vfujywLdW_s?si=UBQ_bMdAELaWcbNC\" width=\"560\" height=\"315\" 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\/Borda+Count.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBorda Count\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm12858\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=12858&theme=lumen&iframe_resize_id=ohm12858&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>What\u2019s Wrong with Borda Count?<\/h3>\n<p>You might have already noticed one potential flaw of the Borda count from the previous example. In that example, Seattle had a majority of first-choice votes, yet lost the election! This seems odd, and prompts our next fairness criterion:<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>majority criterion<\/h3>\n<p>If a choice has a majority of first-place votes, that choice should be the winner.<\/p>\n<\/div>\n<\/section>\n<p>The election from the previous example using the Borda count violates the majority criterion. Notice also that this automatically means that the Condorcet Criterion will also be violated, as Seattle would have been preferred by [latex]51\\%[\/latex] of voters in any head-to-head comparison. Borda count is sometimes described as a consensus-based voting system, since it can sometimes choose a more broadly acceptable option over the one with majority support. In the example above, Tacoma is probably the best compromise location. This is a different approach than plurality and instant runoff voting that focus on first-choice votes; Borda count considers every voter\u2019s entire ranking to determine the outcome. Because of this consensus behavior, Borda count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy.<\/p>\n","protected":false},"author":15,"menu_order":6,"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\/8840"}],"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":17,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8840\/revisions"}],"predecessor-version":[{"id":14869,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8840\/revisions\/14869"}],"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\/8840\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8840"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8840"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8840"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}