{"id":8309,"date":"2023-09-29T14:33:31","date_gmt":"2023-09-29T14:33:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8309"},"modified":"2024-10-18T20:58:41","modified_gmt":"2024-10-18T20:58:41","slug":"weighted-voting-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/weighted-voting-learn-it-1\/","title":{"raw":"Weighted Voting: Learn It 1","rendered":"Weighted Voting: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Define weighted voting and distinguish it from equal voting<\/li>\r\n\t<li>Apply the Banzhaf power index and the Shapley-Shubik power Index to assess the relative power of individual voters in a weighted voting situation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Weighted Voting<\/h2>\r\n<p>In a corporate shareholders meeting, each shareholders\u2019 vote counts proportional to the amount of shares they own. An individual with one share gets the equivalent of one vote, while someone with [latex]100[\/latex] shares gets the equivalent of [latex]100[\/latex] votes. This is called <strong>weighted voting<\/strong>, where each vote has some weight attached to it. Weighted voting is sometimes used to vote on candidates, but more commonly to decide \u201cyes\u201d or \u201cno\u201d on a proposal, sometimes called a <strong>motion<\/strong>. Weighted voting is applicable in corporate settings, as well as decision making in parliamentary governments and voting in the United Nations Security Council<\/p>\r\n<p>In weighted voting, we are most often interested in the power each voter has in influencing the outcome.<\/p>\r\n<p>We\u2019ll begin with some basic vocabulary for weighted voting systems.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>vocabulary for weighted voting<\/h3>\r\n<ul>\r\n\t<li>Each individual or entity casting a vote is called a <strong>player <\/strong>in the election. They\u2019re often notated as [latex]P_1,P_2,P_3,...,P_N[\/latex] where [latex]N[\/latex] is the total number of voters.<\/li>\r\n\t<li>Each player is given a <strong>weight<\/strong>, which usually represents how many votes they get.<\/li>\r\n\t<li>The <strong>quota <\/strong>is the minimum weight needed for the votes or weight needed for the proposal to be approved.<\/li>\r\n\t<li>A weighted voting system will often be represented in a shorthand form:\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n<center>[latex][q: w_1, w_2, w_3, ... , w_N][\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n\r\nIn this form, [latex]q[\/latex] is the quota, [latex]w_1[\/latex] is the weight for player 1, and so on.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>In a small company, there are [latex]4[\/latex] shareholders. Mr. Smith has a [latex]30\\%[\/latex] ownership stake in the company, Mr. Garcia has a [latex]25\\%[\/latex] stake, Mrs. Hughes has a [latex]25\\%[\/latex] stake, and Mrs. Lee has a [latex]20\\%[\/latex] stake. They are trying to decide whether to open a new location. The company by-laws state that more than [latex]50\\%[\/latex] of the ownership has to approve any decision like this. This could be represented by the weighted voting system:<\/p>\r\n<p style=\"text-align: center;\">[latex][51: 30, 25, 25, 20][\/latex]<\/p>\r\n<p>Here we have treated the percentage ownership as votes, so Mr. Smith gets the equivalent of [latex]30[\/latex] votes, having a [latex]30\\%[\/latex] ownership stake. Since more than [latex]50\\%[\/latex] is required to approve the decision, the quota is [latex]51[\/latex], the smallest whole number over [latex]50[\/latex].<\/p>\r\n<\/section>\r\n<p>In order to have a meaningful weighted voting system, it is necessary to put some limits on the quota.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>limits on the quota<\/h3>\r\n<p>The quota must be more than [latex]\\frac{1}{2}[\/latex] the total number of votes.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The quota can\u2019t be larger than the total number of votes.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>You may be wondering why these limitations are set. Let's consider the voting system: [latex][q; 3, 2, 1][\/latex]<\/p>\r\n<p>Here there are [latex]6[\/latex] total votes. If the quota was set at only [latex]3[\/latex], then player 1 could vote yes, players 2 and 3 could vote no, and both would reach quota, which doesn\u2019t lead to a decision being made. In order for only one decision to reach quota at a time, the quota must be at least half the total number of votes. If the quota was set to [latex]7[\/latex], then no group of voters could ever reach quota, and no decision can be made, so it doesn\u2019t make sense for the quota to be larger than the total number of voters.<\/p>\r\n<section class=\"textbox example\">\r\n<p>In a committee there are four representatives from the management and three representatives from the workers\u2019 union. For a proposal to pass, four of the members must support it, including at least one member of the union. Find a voting system that can represent this situation. [reveal-answer q=\"4331\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4331\"]<\/p>\r\n<p>If we represent the players as [latex]M_{1}, M_{2}, M_{3}, M_{4}, U_{I}, U_{2}, U_{3}[\/latex] then we may be tempted to set up a system like [latex][4: 1, 1, 1, 1, 1, 1, 1][\/latex]. While this system would meet the first requirement that four members must support a proposal for it to pass, this does not satisfy the requirement that at least one member of the union must support it.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13265[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Define weighted voting and distinguish it from equal voting<\/li>\n<li>Apply the Banzhaf power index and the Shapley-Shubik power Index to assess the relative power of individual voters in a weighted voting situation<\/li>\n<\/ul>\n<\/section>\n<h2>Weighted Voting<\/h2>\n<p>In a corporate shareholders meeting, each shareholders\u2019 vote counts proportional to the amount of shares they own. An individual with one share gets the equivalent of one vote, while someone with [latex]100[\/latex] shares gets the equivalent of [latex]100[\/latex] votes. This is called <strong>weighted voting<\/strong>, where each vote has some weight attached to it. Weighted voting is sometimes used to vote on candidates, but more commonly to decide \u201cyes\u201d or \u201cno\u201d on a proposal, sometimes called a <strong>motion<\/strong>. Weighted voting is applicable in corporate settings, as well as decision making in parliamentary governments and voting in the United Nations Security Council<\/p>\n<p>In weighted voting, we are most often interested in the power each voter has in influencing the outcome.<\/p>\n<p>We\u2019ll begin with some basic vocabulary for weighted voting systems.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>vocabulary for weighted voting<\/h3>\n<ul>\n<li>Each individual or entity casting a vote is called a <strong>player <\/strong>in the election. They\u2019re often notated as [latex]P_1,P_2,P_3,...,P_N[\/latex] where [latex]N[\/latex] is the total number of voters.<\/li>\n<li>Each player is given a <strong>weight<\/strong>, which usually represents how many votes they get.<\/li>\n<li>The <strong>quota <\/strong>is the minimum weight needed for the votes or weight needed for the proposal to be approved.<\/li>\n<li>A weighted voting system will often be represented in a shorthand form:\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex][q: w_1, w_2, w_3, ... , w_N][\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>In this form, [latex]q[\/latex] is the quota, [latex]w_1[\/latex] is the weight for player 1, and so on.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>In a small company, there are [latex]4[\/latex] shareholders. Mr. Smith has a [latex]30\\%[\/latex] ownership stake in the company, Mr. Garcia has a [latex]25\\%[\/latex] stake, Mrs. Hughes has a [latex]25\\%[\/latex] stake, and Mrs. Lee has a [latex]20\\%[\/latex] stake. They are trying to decide whether to open a new location. The company by-laws state that more than [latex]50\\%[\/latex] of the ownership has to approve any decision like this. This could be represented by the weighted voting system:<\/p>\n<p style=\"text-align: center;\">[latex][51: 30, 25, 25, 20][\/latex]<\/p>\n<p>Here we have treated the percentage ownership as votes, so Mr. Smith gets the equivalent of [latex]30[\/latex] votes, having a [latex]30\\%[\/latex] ownership stake. Since more than [latex]50\\%[\/latex] is required to approve the decision, the quota is [latex]51[\/latex], the smallest whole number over [latex]50[\/latex].<\/p>\n<\/section>\n<p>In order to have a meaningful weighted voting system, it is necessary to put some limits on the quota.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>limits on the quota<\/h3>\n<p>The quota must be more than [latex]\\frac{1}{2}[\/latex] the total number of votes.<\/p>\n<p>&nbsp;<\/p>\n<p>The quota can\u2019t be larger than the total number of votes.<\/p>\n<\/div>\n<\/section>\n<p>You may be wondering why these limitations are set. Let&#8217;s consider the voting system: [latex][q; 3, 2, 1][\/latex]<\/p>\n<p>Here there are [latex]6[\/latex] total votes. If the quota was set at only [latex]3[\/latex], then player 1 could vote yes, players 2 and 3 could vote no, and both would reach quota, which doesn\u2019t lead to a decision being made. In order for only one decision to reach quota at a time, the quota must be at least half the total number of votes. If the quota was set to [latex]7[\/latex], then no group of voters could ever reach quota, and no decision can be made, so it doesn\u2019t make sense for the quota to be larger than the total number of voters.<\/p>\n<section class=\"textbox example\">\n<p>In a committee there are four representatives from the management and three representatives from the workers\u2019 union. For a proposal to pass, four of the members must support it, including at least one member of the union. Find a voting system that can represent this situation. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4331\">Show Solution<\/button><\/p>\n<div id=\"q4331\" class=\"hidden-answer\" style=\"display: none\">\n<p>If we represent the players as [latex]M_{1}, M_{2}, M_{3}, M_{4}, U_{I}, U_{2}, U_{3}[\/latex] then we may be tempted to set up a system like [latex][4: 1, 1, 1, 1, 1, 1, 1][\/latex]. While this system would meet the first requirement that four members must support a proposal for it to pass, this does not satisfy the requirement that at least one member of the union must support it.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13265\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13265&theme=lumen&iframe_resize_id=ohm13265&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Math in Society (Lippman)\",\"author\":\"David Lippman\",\"organization\":\"LibreTexts Mathematics\",\"url\":\"https:\/\/math.libretexts.org\/Bookshelves\/Applied_Mathematics\/Math_in_Society_(Lippman)\/03%3A_Weighted_Voting\/3.02%3A_Beginnings\",\"project\":\"3.2: Beginnings\",\"license\":\"cc-by-sa\",\"license_terms\":\"Access for free at https:\/\/math.libretexts.org\/Bookshelves\/Applied_Mathematics\/Math_in_Society_(Lippman)\"}]","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":"cc-attribution","description":"Math in Society (Lippman)","author":"David Lippman","organization":"LibreTexts Mathematics","url":"https:\/\/math.libretexts.org\/Bookshelves\/Applied_Mathematics\/Math_in_Society_(Lippman)\/03%3A_Weighted_Voting\/3.02%3A_Beginnings","project":"3.2: Beginnings","license":"cc-by-sa","license_terms":"Access for free at https:\/\/math.libretexts.org\/Bookshelves\/Applied_Mathematics\/Math_in_Society_(Lippman)"}],"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\/8309"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8309\/revisions"}],"predecessor-version":[{"id":12810,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8309\/revisions\/12810"}],"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\/8309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}