{"id":8962,"date":"2023-10-12T17:43:55","date_gmt":"2023-10-12T17:43:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8962"},"modified":"2024-10-18T20:58:41","modified_gmt":"2024-10-18T20:58:41","slug":"weighted-voting-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/weighted-voting-learn-it-2\/","title":{"raw":"Weighted Voting: Learn It 2","rendered":"Weighted Voting: Learn It 2"},"content":{"raw":"<h2>A Look at Power<\/h2>\r\n<p>Consider the voting system [latex][10:11,3,2][\/latex]. Notice that in this system, player 1 can reach quota without the support of any other player. When this happens, we say that player 1 is a <strong>dictator<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>dictator<\/h3>\r\n\r\n\r\nA player will be a <strong>dictator <\/strong>if their weight is equal to or greater than the quota. The dictator can also block any proposal from passing; the other players cannot reach quota without the dictator.<\/div>\r\n<\/section>\r\n<p>In the voting system [latex][8:6,3,2][\/latex], no player is a dictator. However, in this system, the quota can only be reached if player 1 is in support of the proposal; player 2 and 3 cannot reach quota without player 1\u2019s support. In this case, player 1 is said to have <strong>veto power<\/strong>. Notice that player 1 is not a dictator, since player 1 would still need player 2 or 3\u2019s support to reach quota.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>veto power<\/h3>\r\n\r\n\r\nA player has <strong>veto power<\/strong> if their support is necessary for the quota to be reached. It is possible for more than one player to have veto power, or for no player to have veto power.<\/div>\r\n<\/section>\r\n<p>With the system [latex][10:7,6,2][\/latex], player 3 is said to be a <strong>dummy<\/strong>, meaning they have no influence in the outcome. The only way the quota can be met is with the support of both players 1 and 2 (both of which would have veto power here); the vote of player 3 cannot affect the outcome.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>dummy<\/h3>\r\n\r\n\r\nA player is a <strong>dummy <\/strong>if their vote is never essential for a group to reach quota.<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">In the voting system [latex][16:7,6,3,3,2][\/latex], are any players dictators? Do any have veto power? Are any dummies? [reveal-answer q=\"4331\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4331\"]\r\n\r\n\r\n<p>No player can reach quota alone, so there are no dictators.<\/p>\r\n<p>Without player 1, the rest of the players\u2019 weights add to [latex]14[\/latex], which doesn\u2019t reach quota, so player 1 has veto power. Likewise, without player 2, the rest of the players\u2019 weights add to [latex]15[\/latex], which doesn\u2019t reach quota, so player 2 also has veto power.<\/p>\r\n<p>Since player 1 and 2 can reach quota with either player 3 or player 4\u2019s support, neither player 3 or player 4 have veto power. However they cannot reach quota with player 5\u2019s support alone, so player 5 has no influence on the outcome and is a dummy.<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>To better define power, we need to introduce the idea of a <strong>coalition<\/strong>. A coalition is a group of players voting the same way. In the example above, [latex]\\{P1,P2,P4\\}[\/latex] would represent the coalition of players 1, 2 and 4. This coalition has a combined weight of [latex]7+6+3=16[\/latex], which meets quota, so this would be a winning coalition.<\/p>\r\n<p>A player is said to be <strong>critical <\/strong>in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition. In the coalition [latex]\\{P1,P2,P4\\}[\/latex], every player is critical. In the coalition [latex]{P3,P4,P5}[\/latex], no player is critical, since it wasn\u2019t a winning coalition to begin with. In the coalition [latex]\\{P1,P2,P3,P4,P5\\}[\/latex], only players 1 and 2 are critical; any other player could leave the coalition and it would still meet quota.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>coalition and critical players<\/h3>\r\n<p>A <strong>coalition <\/strong>is any group of players voting the same way.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A coalition is a <strong>winning coalition<\/strong> if the coalition has enough weight to meet quota.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A player is <strong>critical<\/strong> in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">In the Scottish Parliament in 2009 there were [latex]5[\/latex] political parties: [latex]47[\/latex] representatives for the Scottish National Party, [latex]46[\/latex] for the Labour Party, [latex]17[\/latex] for the Conservative Party, [latex]16[\/latex] for the Liberal Democrats, and [latex]2[\/latex] for the Scottish Green Party. Typically all representatives from a party vote as a block, so the parliament can be treated like the weighted voting system:\r\n\r\n<p style=\"text-align: center;\">[latex][65: 47, 46, 17, 16, 2][\/latex]<\/p>\r\n<p>Consider the coalition [latex]\\{P1,P3,P4\\}[\/latex]. No two players alone could meet the quota, so all three players are critical in this coalition.<\/p>\r\n<p>In the coalition [latex]\\{P1,P3,P4,P5\\}[\/latex], any player except [latex]P1[\/latex] could leave the coalition and it would still meet quota, so only [latex]P1[\/latex] is critical in this coalition.<\/p>\r\n<\/section>\r\n<p>Notice that a player with veto power will be critical in every winning coalition, since removing their support would prevent a proposal from passing.<\/p>\r\n<p>Likewise, a dummy will never be critical, since their support will never change a losing coalition to a winning one.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>dictators, veto, dummies and critical players<\/h3>\r\n<p>A player is a <strong>dictator <\/strong>if the single-player coalition containing them is a winning coalition.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A player has<strong> veto power<\/strong> if they are critical in every winning coalition.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A player is a <strong>dummy<\/strong> if they are not critical in any winning coalition.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13266[\/ohm2_question]<\/section>","rendered":"<h2>A Look at Power<\/h2>\n<p>Consider the voting system [latex][10:11,3,2][\/latex]. Notice that in this system, player 1 can reach quota without the support of any other player. When this happens, we say that player 1 is a <strong>dictator<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>dictator<\/h3>\n<p>A player will be a <strong>dictator <\/strong>if their weight is equal to or greater than the quota. The dictator can also block any proposal from passing; the other players cannot reach quota without the dictator.<\/div>\n<\/section>\n<p>In the voting system [latex][8:6,3,2][\/latex], no player is a dictator. However, in this system, the quota can only be reached if player 1 is in support of the proposal; player 2 and 3 cannot reach quota without player 1\u2019s support. In this case, player 1 is said to have <strong>veto power<\/strong>. Notice that player 1 is not a dictator, since player 1 would still need player 2 or 3\u2019s support to reach quota.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>veto power<\/h3>\n<p>A player has <strong>veto power<\/strong> if their support is necessary for the quota to be reached. It is possible for more than one player to have veto power, or for no player to have veto power.<\/div>\n<\/section>\n<p>With the system [latex][10:7,6,2][\/latex], player 3 is said to be a <strong>dummy<\/strong>, meaning they have no influence in the outcome. The only way the quota can be met is with the support of both players 1 and 2 (both of which would have veto power here); the vote of player 3 cannot affect the outcome.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>dummy<\/h3>\n<p>A player is a <strong>dummy <\/strong>if their vote is never essential for a group to reach quota.<\/div>\n<\/section>\n<section class=\"textbox example\">In the voting system [latex][16:7,6,3,3,2][\/latex], are any players dictators? Do any have veto power? Are any dummies? <\/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>No player can reach quota alone, so there are no dictators.<\/p>\n<p>Without player 1, the rest of the players\u2019 weights add to [latex]14[\/latex], which doesn\u2019t reach quota, so player 1 has veto power. Likewise, without player 2, the rest of the players\u2019 weights add to [latex]15[\/latex], which doesn\u2019t reach quota, so player 2 also has veto power.<\/p>\n<p>Since player 1 and 2 can reach quota with either player 3 or player 4\u2019s support, neither player 3 or player 4 have veto power. However they cannot reach quota with player 5\u2019s support alone, so player 5 has no influence on the outcome and is a dummy.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>To better define power, we need to introduce the idea of a <strong>coalition<\/strong>. A coalition is a group of players voting the same way. In the example above, [latex]\\{P1,P2,P4\\}[\/latex] would represent the coalition of players 1, 2 and 4. This coalition has a combined weight of [latex]7+6+3=16[\/latex], which meets quota, so this would be a winning coalition.<\/p>\n<p>A player is said to be <strong>critical <\/strong>in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition. In the coalition [latex]\\{P1,P2,P4\\}[\/latex], every player is critical. In the coalition [latex]{P3,P4,P5}[\/latex], no player is critical, since it wasn\u2019t a winning coalition to begin with. In the coalition [latex]\\{P1,P2,P3,P4,P5\\}[\/latex], only players 1 and 2 are critical; any other player could leave the coalition and it would still meet quota.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>coalition and critical players<\/h3>\n<p>A <strong>coalition <\/strong>is any group of players voting the same way.<\/p>\n<p>&nbsp;<\/p>\n<p>A coalition is a <strong>winning coalition<\/strong> if the coalition has enough weight to meet quota.<\/p>\n<p>&nbsp;<\/p>\n<p>A player is <strong>critical<\/strong> in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">In the Scottish Parliament in 2009 there were [latex]5[\/latex] political parties: [latex]47[\/latex] representatives for the Scottish National Party, [latex]46[\/latex] for the Labour Party, [latex]17[\/latex] for the Conservative Party, [latex]16[\/latex] for the Liberal Democrats, and [latex]2[\/latex] for the Scottish Green Party. Typically all representatives from a party vote as a block, so the parliament can be treated like the weighted voting system:<\/p>\n<p style=\"text-align: center;\">[latex][65: 47, 46, 17, 16, 2][\/latex]<\/p>\n<p>Consider the coalition [latex]\\{P1,P3,P4\\}[\/latex]. No two players alone could meet the quota, so all three players are critical in this coalition.<\/p>\n<p>In the coalition [latex]\\{P1,P3,P4,P5\\}[\/latex], any player except [latex]P1[\/latex] could leave the coalition and it would still meet quota, so only [latex]P1[\/latex] is critical in this coalition.<\/p>\n<\/section>\n<p>Notice that a player with veto power will be critical in every winning coalition, since removing their support would prevent a proposal from passing.<\/p>\n<p>Likewise, a dummy will never be critical, since their support will never change a losing coalition to a winning one.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>dictators, veto, dummies and critical players<\/h3>\n<p>A player is a <strong>dictator <\/strong>if the single-player coalition containing them is a winning coalition.<\/p>\n<p>&nbsp;<\/p>\n<p>A player has<strong> veto power<\/strong> if they are critical in every winning coalition.<\/p>\n<p>&nbsp;<\/p>\n<p>A player is a <strong>dummy<\/strong> if they are not critical in any winning coalition.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13266\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13266&theme=lumen&iframe_resize_id=ohm13266&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":24,"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\/3.03%3A_A_Look_at_Power\",\"project\":\"3.3: A Look at Power\",\"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\/3.03%3A_A_Look_at_Power","project":"3.3: A Look at Power","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\/8962"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8962\/revisions"}],"predecessor-version":[{"id":12811,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8962\/revisions\/12811"}],"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\/8962\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8962"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8962"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8962"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8962"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}