{"id":1580,"date":"2017-02-16T18:41:33","date_gmt":"2017-02-16T18:41:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1580"},"modified":"2021-02-18T02:29:12","modified_gmt":"2021-02-18T02:29:12","slug":"borda-count","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/borda-count\/","title":{"raw":"Borda Count","rendered":"Borda Count"},"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 a Borda count<\/li>\r\n \t<li>Evaluate the fairness of an election determined using a Borda count<\/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\nBorda Count is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.\r\n<div class=\"textbox\">\r\n<h3>Borda Count<\/h3>\r\nIn this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 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.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA 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.\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/16183951\/Puget-Sound.png\"><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\" \/><\/a>\r\n\r\nThe votes for where to hold the conference were:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>51<\/td>\r\n<td>25<\/td>\r\n<td>10<\/td>\r\n<td>14<\/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\nUse the Borda count method to determine the winning town for the conference.\r\n\r\n[reveal-answer q=\"75557\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"75557\"]\r\n\r\nIn each of the 51 ballots ranking Seattle first, Puyallup will be given 1 point, Olympia 2 points, Tacoma 3 points, and Seattle 4 points. Multiplying the points per vote times the number of votes allows us to calculate points awarded\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>51<\/td>\r\n<td>25<\/td>\r\n<td>10<\/td>\r\n<td>14<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice 4 points<\/td>\r\n<td>Seattle\r\n\r\n[latex]4\\cdot51=204[\/latex]<\/td>\r\n<td>Tacoma\r\n\r\n[latex]4\\cdot25=100[\/latex]<\/td>\r\n<td>Puyallup\r\n\r\n[latex]4\\cot10=40[\/latex]<\/td>\r\n<td>Olympia\r\n\r\n[latex]4\\cdot14=56[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice 3 points<\/td>\r\n<td>Tacoma\r\n\r\n[latex]3\\cdot51=153[\/latex]<\/td>\r\n<td>Puyallup\r\n\r\n[latex]3\\cdot25=75[\/latex]<\/td>\r\n<td>Tacoma\r\n\r\n[latex]3\\cdot10=30[\/latex]<\/td>\r\n<td>Tacoma\r\n\r\n[latex]3\\cdot14=42[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice 2 points<\/td>\r\n<td>Olympia\r\n\r\n[latex]2\\cdot51=102[\/latex]<\/td>\r\n<td>Olympia\r\n\r\n[latex]2\\cdot25=50[\/latex]<\/td>\r\n<td>Olympia\r\n\r\n[latex]2\\cdot10=20[\/latex]<\/td>\r\n<td>Puyallup\r\n\r\n[latex]2\\cdot14=28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice 1 point<\/td>\r\n<td>Puyallup\r\n\r\n[latex]1\\cdot51=51[\/latex]<\/td>\r\n<td>Seattle\r\n\r\n[latex]1\\cdot25=25[\/latex]<\/td>\r\n<td>Seattle\r\n\r\n[latex]1\\cdot10=10[\/latex]<\/td>\r\n<td>Seattle\r\n\r\n[latex]1\\cdot14=14[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAdding up the points:\r\n\r\nSeattle: [latex]204+25+10+14=253[\/latex] points\r\n\r\nTacoma: [latex]153+100+30+42=325[\/latex] points\r\n\r\nPuyallup: [latex]51+75+40+28=194[\/latex] points\r\n\r\nOlympia: [latex]102+50+20+56=228[\/latex] points\r\n\r\nUnder the Borda Count method, Tacoma is the winner of this vote.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is a video showing the example from above.\r\n\r\nhttps:\/\/youtu.be\/vfujywLdW_s?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 Borda Count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate 1 point, the points they would normally get for last place.\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=\"159776\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"159776\"]\r\n\r\nWe give 1 point for 3rd place, 2 points for 2nd place, and 3 points for 1st place. In the columns where voters only listed first place choice, the same number of votes are awarded to each of the other candidates as 3rd place vote multipliers\r\n\r\nG: 132+42+60+140+22+80+39 = 515 points\r\n\r\nM: 88+14+20+210+66+160+39 = 597 points\r\n\r\nB: 44+28+20+70+44+240+117 = 563 points\r\n\r\nMcCarthy would be the winner using Borda Count.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>What\u2019s Wrong with Borda Count?<\/h2>\r\nYou 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:\r\n<div class=\"textbox\">\r\n<h3>Majority Criterion<\/h3>\r\nIf a choice has a majority of first-place votes, that choice should be the winner.\r\n\r\n<\/div>\r\nThe election from the previous example\u00a0using 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 51% of voters in any head-to-head comparison.\r\n\r\nBorda 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.\r\n\r\nBecause 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.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\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<\/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>Borda Count is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.<\/p>\n<div class=\"textbox\">\n<h3>Borda Count<\/h3>\n<p>In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\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.<br \/>\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/16183951\/Puget-Sound.png\"><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\" \/><\/a><\/p>\n<p>The votes for where to hold the conference were:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>51<\/td>\n<td>25<\/td>\n<td>10<\/td>\n<td>14<\/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>Use the Borda count method to determine the winning town for the conference.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q75557\">Show Solution<\/span><\/p>\n<div id=\"q75557\" class=\"hidden-answer\" style=\"display: none\">\n<p>In each of the 51 ballots ranking Seattle first, Puyallup will be given 1 point, Olympia 2 points, Tacoma 3 points, and Seattle 4 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><\/td>\n<td>51<\/td>\n<td>25<\/td>\n<td>10<\/td>\n<td>14<\/td>\n<\/tr>\n<tr>\n<td>1st choice 4 points<\/td>\n<td>Seattle<\/p>\n<p>[latex]4\\cdot51=204[\/latex]<\/td>\n<td>Tacoma<\/p>\n<p>[latex]4\\cdot25=100[\/latex]<\/td>\n<td>Puyallup<\/p>\n<p>[latex]4\\cot10=40[\/latex]<\/td>\n<td>Olympia<\/p>\n<p>[latex]4\\cdot14=56[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2nd choice 3 points<\/td>\n<td>Tacoma<\/p>\n<p>[latex]3\\cdot51=153[\/latex]<\/td>\n<td>Puyallup<\/p>\n<p>[latex]3\\cdot25=75[\/latex]<\/td>\n<td>Tacoma<\/p>\n<p>[latex]3\\cdot10=30[\/latex]<\/td>\n<td>Tacoma<\/p>\n<p>[latex]3\\cdot14=42[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3rd choice 2 points<\/td>\n<td>Olympia<\/p>\n<p>[latex]2\\cdot51=102[\/latex]<\/td>\n<td>Olympia<\/p>\n<p>[latex]2\\cdot25=50[\/latex]<\/td>\n<td>Olympia<\/p>\n<p>[latex]2\\cdot10=20[\/latex]<\/td>\n<td>Puyallup<\/p>\n<p>[latex]2\\cdot14=28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>4th choice 1 point<\/td>\n<td>Puyallup<\/p>\n<p>[latex]1\\cdot51=51[\/latex]<\/td>\n<td>Seattle<\/p>\n<p>[latex]1\\cdot25=25[\/latex]<\/td>\n<td>Seattle<\/p>\n<p>[latex]1\\cdot10=10[\/latex]<\/td>\n<td>Seattle<\/p>\n<p>[latex]1\\cdot14=14[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Adding up the points:<\/p>\n<p>Seattle: [latex]204+25+10+14=253[\/latex] points<\/p>\n<p>Tacoma: [latex]153+100+30+42=325[\/latex] points<\/p>\n<p>Puyallup: [latex]51+75+40+28=194[\/latex] points<\/p>\n<p>Olympia: [latex]102+50+20+56=228[\/latex] points<\/p>\n<p>Under the Borda Count method, Tacoma is the winner of this vote.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here is a video showing the example from above.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Borda Count\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vfujywLdW_s?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 Borda Count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate 1 point, the points they would normally get for last place.<\/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=\"q159776\">Solution<\/span><\/p>\n<div id=\"q159776\" class=\"hidden-answer\" style=\"display: none\">\n<p>We give 1 point for 3rd place, 2 points for 2nd place, and 3 points for 1st place. In the columns where voters only listed first place choice, the same number of votes are awarded to each of the other candidates as 3rd place vote multipliers<\/p>\n<p>G: 132+42+60+140+22+80+39 = 515 points<\/p>\n<p>M: 88+14+20+210+66+160+39 = 597 points<\/p>\n<p>B: 44+28+20+70+44+240+117 = 563 points<\/p>\n<p>McCarthy would be the winner using Borda Count.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>What\u2019s Wrong with Borda Count?<\/h2>\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<div class=\"textbox\">\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<p>The election from the previous example\u00a0using 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 51% of voters in any head-to-head comparison.<\/p>\n<p>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.<\/p>\n<p>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\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-1580\">\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>Borda Count . <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vfujywLdW_s?list=PL1F887D3B8BF7C297\">https:\/\/youtu.be\/vfujywLdW_s?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":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Borda Count \",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/vfujywLdW_s?list=PL1F887D3B8BF7C297\",\"project\":\"Open Course Library\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"ab1c42d4-ea9c-4e19-8ed9-dbd2a5e49ff6","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1580","chapter","type-chapter","status-publish","hentry"],"part":1040,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1580","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1580\/revisions"}],"predecessor-version":[{"id":5364,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1580\/revisions\/5364"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/1040"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1580\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=1580"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1580"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=1580"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=1580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}