{"id":8312,"date":"2023-09-29T14:33:46","date_gmt":"2023-09-29T14:33:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8312"},"modified":"2024-10-18T20:58:43","modified_gmt":"2024-10-18T20:58:43","slug":"weighted-voting-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/weighted-voting-fresh-take\/","title":{"raw":"Weighted Voting: Fresh Take","rendered":"Weighted Voting: Fresh Take"},"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<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Weighted Voting:<\/strong> A system where each vote has a weight attached to it, often proportional to ownership stakes or representation.<\/p>\r\n<p><strong>Player:<\/strong> An individual or entity casting a vote in the election, often notated as [latex]P_1,P_2,P_3,...,P_N[\/latex] where [latex]N[\/latex] is the total number of voters.<\/p>\r\n<p><strong>Weight:<\/strong> The value assigned to each player, representing their voting power.<\/p>\r\n<p><strong>Quota:<\/strong> The minimum weight needed for a proposal to be approved.<\/p>\r\n<p><strong>Shorthand Representation:<\/strong> A compact way to represent a weighted voting system, e.g., [latex][q: w_1, w_2, w_3, ... , w_N][\/latex]<\/p>\r\n\r\nBelow are important things to consider when thinking about weighted voting.\r\n\r\n<ul>\r\n\t<li><strong>Understanding Weight:<\/strong> In a corporate setting, your weight might be directly tied to the number of shares you own. More shares mean more weight.<\/li>\r\n\t<li><strong>Quota Constraints:<\/strong> The quota must be more than half of the total number of votes but can't be larger than the total number of votes. This ensures a decisive outcome.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>A Look at Power<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p>In weighted voting systems, players can have different types of power, such as being a dictator, having veto power, or being a dummy. A <strong>dictator<\/strong> is a player whose weight alone can meet the quota, making them the ultimate decision-maker. <strong>Veto power<\/strong> means a player's support is essential for reaching the quota, but they can't do it alone. A <strong>dummy<\/strong>is a player whose vote doesn't influence the outcome. <strong>Coalitions<\/strong> are groups of players voting the same way, and a player is <strong>critical <\/strong>in a coalition if their departure changes the coalition's status from winning to losing.<\/p>\r\n\r\nBelow are some key terms.\r\n\r\n<ul>\r\n\t<li><strong>Dictator:<\/strong> If a player's weight is equal to or greater than the quota, they are a dictator. They can pass or block any proposal single-handedly.<\/li>\r\n\t<li><strong>Veto Power:<\/strong> A player has veto power if their support is necessary for the quota to be reached. They can't pass a proposal alone but can block one.<\/li>\r\n\t<li><strong>Dummy:<\/strong> A player is a dummy if their vote is never essential for a group to reach the quota. Their presence or absence doesn't affect the outcome.<\/li>\r\n\t<li><strong>Coalition:<\/strong> A coalition is a group of players voting the same way. It's a winning coalition if it has enough weight to meet the quota.<\/li>\r\n\t<li><strong>Critical Players:<\/strong> A player is critical in a coalition if their departure changes it from a winning coalition to a losing one.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>For more information on these terms and the basics of weighted voting, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Iaxblazgb1Y?si=Io-xMZTzBAKOcxe8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Introduction+to+Weighted+Voting.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Weighted Voting\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Calculating Power- Banzhaf Power Index<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p>The <strong>Banzhaf Power Index<\/strong> is a fascinating tool that quantifies the influence of individual players in a weighted voting system. It's not just about the number of votes you have; it's about how crucial your votes are to forming winning coalitions. In essence, this index tells us how often a player's vote is a game-changer in reaching a decision.<\/p>\r\n\r\nBelow are important things to consider when thinking about the Banzhaf Power Index.\r\n\r\n<ul>\r\n\t<li><strong>Listing Winning Coalitions:<\/strong> The first step in calculating the Banzhaf Power Index is to list all the possible winning coalitions. A coalition is a group of players who together have enough votes to win.<\/li>\r\n\t<li><strong>Identifying Critical Players:<\/strong> In each coalition, figure out which players are \"critical,\" meaning the coalition would lose without their votes.<\/li>\r\n\t<li><strong>Counting Critical Appearances:<\/strong> Tally up how many times each player is critical across all coalitions.<\/li>\r\n\t<li><strong>Calculating the Index:<\/strong> Convert the counts to fractions or percentages by dividing the number of times a player is critical by the total number of times any player is critical.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Find the Banzhaf power index for the voting system [latex][8:6,3,2][\/latex]. [reveal-answer q=\"4339\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4339\"]\r\n\r\n<p>We start by listing all winning coalitions. If you aren\u2019t sure how to do this, you can list all coalitions, then eliminate the non-winning coalitions. No player is a dictator, so we\u2019ll only consider two and three player coalitions.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\{P_1,P_2\\}[\/latex]: Total weight: [latex]9[\/latex]. Meets quota. [latex]\\{P_1,P_3\\}[\/latex]: Total weight: [latex]8[\/latex]. Meets quota. [latex]\\{P_2,P_3\\}[\/latex]: Total weight: [latex]5[\/latex]. Does not meet quota. [latex]\\{P_1,P_2,P_3\\}[\/latex]: Total weight: [latex]11[\/latex]. Meets quota.<\/p>\r\n<p>Next we determine which players are critical in each winning coalition. In the winning two-player coalitions, both players are critical since no player can meet quota alone. Underlining the critical players to make it easier to count:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\underline{P}_{1}, \\underline{P}_{2}\\right\\}[\/latex] [latex]\\left\\{\\underline{P}_{1}, \\underline{P}_{3}\\right\\}[\/latex]<\/p>\r\n<p>In the three-person coalition, either [latex]P_2[\/latex] or [latex]P_3[\/latex] could leave the coalition and the remaining players could still meet quota, so neither is critical. If [latex]P_1[\/latex] were to leave, the remaining players could not reach quota, so [latex]P_1[\/latex] is critical.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\underline{P}_{1}, P_{2}, P_{3}\\right\\}[\/latex]<\/p>\r\n<p>Altogether, [latex]P_1[\/latex] is critical [latex]3[\/latex] times, [latex]P_2[\/latex] is critical [latex]1[\/latex] time, and [latex]P_3[\/latex] is critical [latex]1[\/latex] time.<\/p>\r\n<p>Converting to percents:<\/p>\r\n<p style=\"text-align: center;\">[latex]P_{1}=3 \/ 5=60 \\%[\/latex] [latex]P_{2}=1 \/ 5=20 \\%[\/latex] [latex]P_{3}=1 \/ 5=20 \\%[\/latex] [\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Find the Banzhaf power index for the voting system [latex][36: 20, 17, 16, 3][\/latex]. [reveal-answer q=\"4338\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4338\"]\r\n\r\n<p>The voting system tells us that the quota is [latex]36[\/latex], that Player 1 has [latex]20[\/latex] votes (or equivalently, has a weight of [latex]20[\/latex]), Player 2 has [latex]17[\/latex] votes, Player 3 has [latex]16[\/latex] votes, and Player 4 has [latex]3[\/latex] votes.<\/p>\r\n<p>A coalition is any group of one or more players. What we're looking for is winning coalitions - coalitions whose combined votes (weights) add to up to the quota or more. So the coalition [latex]\\{P3,P4\\}[\/latex] is not a winning coalition because the combined weight is [latex]16+3=19[\/latex], which is below the quota.<\/p>\r\n<p>So we look at each possible combination of players and identify the winning ones:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array} {ll} {\\{\\mathrm{P}_1, \\mathrm{P}_2\\}(\\text { weight }: 37)} &amp; {\\{\\mathrm{P}_1, \\mathrm{P}_3\\} \\text { (weight: } 36)} \\\\ {\\{\\mathrm{P}_1, \\mathrm{P}_2, \\mathrm{P}_3\\} \\text { (weight: } 53)} &amp; {\\{\\mathrm{P}_1, \\mathrm{P}_2, \\mathrm{P}_4\\} \\text { (weight: } 40)} \\\\ {\\{\\mathrm{P}_1, \\mathrm{P}_3, \\mathrm{P}_4\\} \\text { (weight: } 39)} &amp; {\\{\\mathrm{P}_1, \\mathrm{P}_2, \\mathrm{P}_3, \\mathrm{P}_4\\} \\text { (weight: } 56)} \\\\ {\\{\\mathrm{P}_2, \\mathrm{P}_3, \\mathrm{P}_4\\}(\\text { weight: } 36)} \\end{array}[\/latex]<\/p>\r\n<p>From the list of winning coalitions, we can see that each player is part of certain winning coalitions. The Banzhaf power index measures the ability of each player to change a losing coalition into a winning one by joining it. In other words, it counts the number of times a player is a \"critical\" player in a winning coalition.<\/p>\r\n<p>To find the Banzhaf power index, we need to identify the critical players in each winning coalition:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} {P_1, P_2} &amp; \\text{Critical:} P_1, P_2 \\\\ {P_1, P_3} &amp; \\text{Critical:} P_1, P_3 \\\\ {P_1, P_2, P_3} &amp; \\text{Critical: None} \\\\ {P_1, P_2, P_4} &amp; \\text{Critical:} P_4 \\\\ {P_1, P_3, P_4} &amp; \\text{Critical:} P_4 \\\\ {P_1, P_2, P_3, P_4} &amp; \\text{Critical: None} \\\\ {P_2, P_3, P_4} &amp; \\text{Critical:} P_2, P_3, P_4 \\end{array}[\/latex]<\/p>\r\n<p>Now, we count the number of times each player is critical:<\/p>\r\n<p style=\"text-align: center;\">[latex]P_1[\/latex] is critical [latex]2[\/latex] times [latex]P_2[\/latex] is critical [latex]3[\/latex] times [latex]P_3[\/latex] is critical [latex]3[\/latex] times [latex]P_4[\/latex] is critical [latex]3[\/latex] times<\/p>\r\n<p>The Banzhaf power index is then calculated by taking the number of times each player is critical and dividing it by the total number of critical positions:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Total Critical Positions} = 2 + 3 + 3 + 3 = 11[\/latex] [latex]\\text{Banzhaf Power Index for} P_1 = \\frac{2}{11}[\/latex] [latex]\\text{Banzhaf Power Index for} P_2 = \\frac{3}{11}[\/latex] [latex]\\text{Banzhaf Power Index for} P_3 = \\frac{3}{11}[\/latex] [latex]\\text{Banzhaf Power Index for} P_4 = \\frac{3}{11}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>For more information on the Banzhaf power index, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/sdWgGzetdWI?si=Tyaka0AJAJT7-VyL\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Weighted+Voting_+The+Banzhaf+Power+Index.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWeighted Voting: The Banzhaf Power Index\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Calculating Power- Shapley-Shubik Power Index<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p>In this section, we're diving into the <strong>Shapley-Shubik Power Index<\/strong>, a method that gives us a nuanced understanding of power dynamics in weighted voting systems. Unlike traditional coalitions where the order of players doesn't matter, this index considers <strong>sequential coalitions<\/strong>\u2014the order in which players join a coalition. The key player in these coalitions is the <strong>pivotal player<\/strong>, the one who turns a losing coalition into a winning one. The index is calculated by listing all sequential coalitions, identifying the pivotal player in each, and then converting these counts into fractions or decimals.<\/p>\r\n\r\nBelow are important things to consider when thinking about the Shapley-Shubik Power Index.\r\n\r\n<ul>\r\n\t<li><strong>Sequential Coalition:<\/strong> A list of players in the order they joined the coalition. Notated with angle brackets like [latex]&lt; P_2,P_1,P_3&gt;[\/latex]<\/li>\r\n\t<li><strong>Pivotal Player:<\/strong> The player who changes the status of a coalition from losing to winning. There can only be one pivotal player in any sequential coalition.<\/li>\r\n\t<li><strong>Factorial ([latex]N![\/latex]):<\/strong> The product of all positive whole numbers up to [latex]N[\/latex]. It gives the total number of sequential coalitions for [latex]N[\/latex] players.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Find the Shapley-Shubik power index for the weighted voting system [latex][36: 20, 17, 15][\/latex]. [reveal-answer q=\"4337\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4337\"]\r\n\r\n<p>Listing all sequential coalitions and identifying the pivotal player:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array} {lll} {&lt; P_{1}, \\underline{P}_{2}, P_{3} &gt;} &amp; {&lt; P_{1}, P_{3}, \\underline{P}_{2} &gt;} &amp; {&lt; P_{2}, \\underline{P}_{1}, P_{3} &gt;} \\\\ {&lt; P_{2}, P_{3}, \\underline{P}_{1} &gt;} &amp; {&lt; P_{3}, P_{2}, \\underline{P}_{1} &gt;} &amp; {&lt; P_{3}, P_{1}, \\underline{P}_{2} &gt;} \\end{array}[\/latex]<\/p>\r\n<p>[latex]P_1[\/latex] is pivotal [latex]3[\/latex] times, [latex]P_2[\/latex] is pivotal [latex]3[\/latex] times, and [latex]P_3[\/latex] is pivotal [latex]0[\/latex] times.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{|l|l|l|} \\hline \\textbf { Player } &amp; \\textbf { Times pivotal } &amp; \\textbf { Power index } \\\\ \\hline P_{1} &amp; 3 &amp; 3 \/ 6=50 \\% \\\\ \\hline P_{2} &amp; 3 &amp; 3 \/ 6=50 \\% \\\\ \\hline P_{3} &amp; 0 &amp; 0 \/ 6=0 \\% \\\\ \\hline \\end{array} [\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>For more information on the Shapley-Shubik power index, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6T7g4AyMIm0?si=2XlO0nznmgnlm1sZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Weighted+Voting_+The+Shapley-Shubik+Power+Index.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWeighted Voting: The Shapley-Shubik Power Index\u201d here (opens in new window).<\/a><\/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<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Weighted Voting:<\/strong> A system where each vote has a weight attached to it, often proportional to ownership stakes or representation.<\/p>\n<p><strong>Player:<\/strong> An individual or entity casting a vote in the election, often notated as [latex]P_1,P_2,P_3,...,P_N[\/latex] where [latex]N[\/latex] is the total number of voters.<\/p>\n<p><strong>Weight:<\/strong> The value assigned to each player, representing their voting power.<\/p>\n<p><strong>Quota:<\/strong> The minimum weight needed for a proposal to be approved.<\/p>\n<p><strong>Shorthand Representation:<\/strong> A compact way to represent a weighted voting system, e.g., [latex][q: w_1, w_2, w_3, ... , w_N][\/latex]<\/p>\n<p>Below are important things to consider when thinking about weighted voting.<\/p>\n<ul>\n<li><strong>Understanding Weight:<\/strong> In a corporate setting, your weight might be directly tied to the number of shares you own. More shares mean more weight.<\/li>\n<li><strong>Quota Constraints:<\/strong> The quota must be more than half of the total number of votes but can&#8217;t be larger than the total number of votes. This ensures a decisive outcome.<\/li>\n<\/ul>\n<\/div>\n<h2>A Look at Power<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p>In weighted voting systems, players can have different types of power, such as being a dictator, having veto power, or being a dummy. A <strong>dictator<\/strong> is a player whose weight alone can meet the quota, making them the ultimate decision-maker. <strong>Veto power<\/strong> means a player&#8217;s support is essential for reaching the quota, but they can&#8217;t do it alone. A <strong>dummy<\/strong>is a player whose vote doesn&#8217;t influence the outcome. <strong>Coalitions<\/strong> are groups of players voting the same way, and a player is <strong>critical <\/strong>in a coalition if their departure changes the coalition&#8217;s status from winning to losing.<\/p>\n<p>Below are some key terms.<\/p>\n<ul>\n<li><strong>Dictator:<\/strong> If a player&#8217;s weight is equal to or greater than the quota, they are a dictator. They can pass or block any proposal single-handedly.<\/li>\n<li><strong>Veto Power:<\/strong> A player has veto power if their support is necessary for the quota to be reached. They can&#8217;t pass a proposal alone but can block one.<\/li>\n<li><strong>Dummy:<\/strong> A player is a dummy if their vote is never essential for a group to reach the quota. Their presence or absence doesn&#8217;t affect the outcome.<\/li>\n<li><strong>Coalition:<\/strong> A coalition is a group of players voting the same way. It&#8217;s a winning coalition if it has enough weight to meet the quota.<\/li>\n<li><strong>Critical Players:<\/strong> A player is critical in a coalition if their departure changes it from a winning coalition to a losing one.<\/li>\n<\/ul>\n<\/div>\n<p>For more information on these terms and the basics of weighted voting, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Iaxblazgb1Y?si=Io-xMZTzBAKOcxe8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Introduction+to+Weighted+Voting.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Weighted Voting\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Calculating Power- Banzhaf Power Index<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p>The <strong>Banzhaf Power Index<\/strong> is a fascinating tool that quantifies the influence of individual players in a weighted voting system. It&#8217;s not just about the number of votes you have; it&#8217;s about how crucial your votes are to forming winning coalitions. In essence, this index tells us how often a player&#8217;s vote is a game-changer in reaching a decision.<\/p>\n<p>Below are important things to consider when thinking about the Banzhaf Power Index.<\/p>\n<ul>\n<li><strong>Listing Winning Coalitions:<\/strong> The first step in calculating the Banzhaf Power Index is to list all the possible winning coalitions. A coalition is a group of players who together have enough votes to win.<\/li>\n<li><strong>Identifying Critical Players:<\/strong> In each coalition, figure out which players are &#8220;critical,&#8221; meaning the coalition would lose without their votes.<\/li>\n<li><strong>Counting Critical Appearances:<\/strong> Tally up how many times each player is critical across all coalitions.<\/li>\n<li><strong>Calculating the Index:<\/strong> Convert the counts to fractions or percentages by dividing the number of times a player is critical by the total number of times any player is critical.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Find the Banzhaf power index for the voting system [latex][8:6,3,2][\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4339\">Show Solution<\/button> <\/p>\n<div id=\"q4339\" class=\"hidden-answer\" style=\"display: none\">\n<p>We start by listing all winning coalitions. If you aren\u2019t sure how to do this, you can list all coalitions, then eliminate the non-winning coalitions. No player is a dictator, so we\u2019ll only consider two and three player coalitions.<\/p>\n<p style=\"text-align: center;\">[latex]\\{P_1,P_2\\}[\/latex]: Total weight: [latex]9[\/latex]. Meets quota. [latex]\\{P_1,P_3\\}[\/latex]: Total weight: [latex]8[\/latex]. Meets quota. [latex]\\{P_2,P_3\\}[\/latex]: Total weight: [latex]5[\/latex]. Does not meet quota. [latex]\\{P_1,P_2,P_3\\}[\/latex]: Total weight: [latex]11[\/latex]. Meets quota.<\/p>\n<p>Next we determine which players are critical in each winning coalition. In the winning two-player coalitions, both players are critical since no player can meet quota alone. Underlining the critical players to make it easier to count:<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\underline{P}_{1}, \\underline{P}_{2}\\right\\}[\/latex] [latex]\\left\\{\\underline{P}_{1}, \\underline{P}_{3}\\right\\}[\/latex]<\/p>\n<p>In the three-person coalition, either [latex]P_2[\/latex] or [latex]P_3[\/latex] could leave the coalition and the remaining players could still meet quota, so neither is critical. If [latex]P_1[\/latex] were to leave, the remaining players could not reach quota, so [latex]P_1[\/latex] is critical.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\underline{P}_{1}, P_{2}, P_{3}\\right\\}[\/latex]<\/p>\n<p>Altogether, [latex]P_1[\/latex] is critical [latex]3[\/latex] times, [latex]P_2[\/latex] is critical [latex]1[\/latex] time, and [latex]P_3[\/latex] is critical [latex]1[\/latex] time.<\/p>\n<p>Converting to percents:<\/p>\n<p style=\"text-align: center;\">[latex]P_{1}=3 \/ 5=60 \\%[\/latex] [latex]P_{2}=1 \/ 5=20 \\%[\/latex] [latex]P_{3}=1 \/ 5=20 \\%[\/latex] <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Find the Banzhaf power index for the voting system [latex][36: 20, 17, 16, 3][\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4338\">Show Solution<\/button> <\/p>\n<div id=\"q4338\" class=\"hidden-answer\" style=\"display: none\">\n<p>The voting system tells us that the quota is [latex]36[\/latex], that Player 1 has [latex]20[\/latex] votes (or equivalently, has a weight of [latex]20[\/latex]), Player 2 has [latex]17[\/latex] votes, Player 3 has [latex]16[\/latex] votes, and Player 4 has [latex]3[\/latex] votes.<\/p>\n<p>A coalition is any group of one or more players. What we&#8217;re looking for is winning coalitions &#8211; coalitions whose combined votes (weights) add to up to the quota or more. So the coalition [latex]\\{P3,P4\\}[\/latex] is not a winning coalition because the combined weight is [latex]16+3=19[\/latex], which is below the quota.<\/p>\n<p>So we look at each possible combination of players and identify the winning ones:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array} {ll} {\\{\\mathrm{P}_1, \\mathrm{P}_2\\}(\\text { weight }: 37)} & {\\{\\mathrm{P}_1, \\mathrm{P}_3\\} \\text { (weight: } 36)} \\\\ {\\{\\mathrm{P}_1, \\mathrm{P}_2, \\mathrm{P}_3\\} \\text { (weight: } 53)} & {\\{\\mathrm{P}_1, \\mathrm{P}_2, \\mathrm{P}_4\\} \\text { (weight: } 40)} \\\\ {\\{\\mathrm{P}_1, \\mathrm{P}_3, \\mathrm{P}_4\\} \\text { (weight: } 39)} & {\\{\\mathrm{P}_1, \\mathrm{P}_2, \\mathrm{P}_3, \\mathrm{P}_4\\} \\text { (weight: } 56)} \\\\ {\\{\\mathrm{P}_2, \\mathrm{P}_3, \\mathrm{P}_4\\}(\\text { weight: } 36)} \\end{array}[\/latex]<\/p>\n<p>From the list of winning coalitions, we can see that each player is part of certain winning coalitions. The Banzhaf power index measures the ability of each player to change a losing coalition into a winning one by joining it. In other words, it counts the number of times a player is a &#8220;critical&#8221; player in a winning coalition.<\/p>\n<p>To find the Banzhaf power index, we need to identify the critical players in each winning coalition:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} {P_1, P_2} & \\text{Critical:} P_1, P_2 \\\\ {P_1, P_3} & \\text{Critical:} P_1, P_3 \\\\ {P_1, P_2, P_3} & \\text{Critical: None} \\\\ {P_1, P_2, P_4} & \\text{Critical:} P_4 \\\\ {P_1, P_3, P_4} & \\text{Critical:} P_4 \\\\ {P_1, P_2, P_3, P_4} & \\text{Critical: None} \\\\ {P_2, P_3, P_4} & \\text{Critical:} P_2, P_3, P_4 \\end{array}[\/latex]<\/p>\n<p>Now, we count the number of times each player is critical:<\/p>\n<p style=\"text-align: center;\">[latex]P_1[\/latex] is critical [latex]2[\/latex] times [latex]P_2[\/latex] is critical [latex]3[\/latex] times [latex]P_3[\/latex] is critical [latex]3[\/latex] times [latex]P_4[\/latex] is critical [latex]3[\/latex] times<\/p>\n<p>The Banzhaf power index is then calculated by taking the number of times each player is critical and dividing it by the total number of critical positions:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Total Critical Positions} = 2 + 3 + 3 + 3 = 11[\/latex] [latex]\\text{Banzhaf Power Index for} P_1 = \\frac{2}{11}[\/latex] [latex]\\text{Banzhaf Power Index for} P_2 = \\frac{3}{11}[\/latex] [latex]\\text{Banzhaf Power Index for} P_3 = \\frac{3}{11}[\/latex] [latex]\\text{Banzhaf Power Index for} P_4 = \\frac{3}{11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>For more information on the Banzhaf power index, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/sdWgGzetdWI?si=Tyaka0AJAJT7-VyL\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Weighted+Voting_+The+Banzhaf+Power+Index.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWeighted Voting: The Banzhaf Power Index\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Calculating Power- Shapley-Shubik Power Index<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p>In this section, we&#8217;re diving into the <strong>Shapley-Shubik Power Index<\/strong>, a method that gives us a nuanced understanding of power dynamics in weighted voting systems. Unlike traditional coalitions where the order of players doesn&#8217;t matter, this index considers <strong>sequential coalitions<\/strong>\u2014the order in which players join a coalition. The key player in these coalitions is the <strong>pivotal player<\/strong>, the one who turns a losing coalition into a winning one. The index is calculated by listing all sequential coalitions, identifying the pivotal player in each, and then converting these counts into fractions or decimals.<\/p>\n<p>Below are important things to consider when thinking about the Shapley-Shubik Power Index.<\/p>\n<ul>\n<li><strong>Sequential Coalition:<\/strong> A list of players in the order they joined the coalition. Notated with angle brackets like [latex]< P_2,P_1,P_3>[\/latex]<\/li>\n<li><strong>Pivotal Player:<\/strong> The player who changes the status of a coalition from losing to winning. There can only be one pivotal player in any sequential coalition.<\/li>\n<li><strong>Factorial ([latex]N![\/latex]):<\/strong> The product of all positive whole numbers up to [latex]N[\/latex]. It gives the total number of sequential coalitions for [latex]N[\/latex] players.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Find the Shapley-Shubik power index for the weighted voting system [latex][36: 20, 17, 15][\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4337\">Show Solution<\/button> <\/p>\n<div id=\"q4337\" class=\"hidden-answer\" style=\"display: none\">\n<p>Listing all sequential coalitions and identifying the pivotal player:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array} {lll} {< P_{1}, \\underline{P}_{2}, P_{3} >} & {< P_{1}, P_{3}, \\underline{P}_{2} >} & {< P_{2}, \\underline{P}_{1}, P_{3} >} \\\\ {< P_{2}, P_{3}, \\underline{P}_{1} >} & {< P_{3}, P_{2}, \\underline{P}_{1} >} & {< P_{3}, P_{1}, \\underline{P}_{2} >} \\end{array}[\/latex]<\/p>\n<p>[latex]P_1[\/latex] is pivotal [latex]3[\/latex] times, [latex]P_2[\/latex] is pivotal [latex]3[\/latex] times, and [latex]P_3[\/latex] is pivotal [latex]0[\/latex] times.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{|l|l|l|} \\hline \\textbf { Player } & \\textbf { Times pivotal } & \\textbf { Power index } \\\\ \\hline P_{1} & 3 & 3 \/ 6=50 \\% \\\\ \\hline P_{2} & 3 & 3 \/ 6=50 \\% \\\\ \\hline P_{3} & 0 & 0 \/ 6=0 \\% \\\\ \\hline \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>For more information on the Shapley-Shubik power index, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6T7g4AyMIm0?si=2XlO0nznmgnlm1sZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Weighted+Voting_+The+Shapley-Shubik+Power+Index.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWeighted Voting: The Shapley-Shubik Power Index\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Weighted Voting\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Iaxblazgb1Y\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Weighted Voting: The Banzhaf Power Index\",\"author\":\"James Sousa 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