Since there are five players, there are 31 coalitions. Previously, the coalition \(\left\{P_{1}, P_{2}\right\}\) and \(\left\{P_{2}, P_{1}\right\}\) would be considered equivalent, since they contain the same players. endobj Player three joining doesnt change the coalitions winning status so it is irrelevant. This is called weighted voting, where each vote has some weight attached to it. Reapportion the previous problem if 37 gold coins are recovered. In Example \(\PageIndex{2}\), some of the weighted voting systems are valid systems. Apply Coombs method to the preference schedules from questions 5 and 6. Using Hamiltons method, apportion the seats based on the 2000 census, then again using the 2010 census. Consider the weighted voting system [q: 10,9,8,8,8,6], Consider the weighted voting system [13: 13, 6, 4, 2], Consider the weighted voting system [11: 9, 6, 3, 1], Consider the weighted voting system [19: 13, 6, 4, 2], Consider the weighted voting system [17: 9, 6, 3, 1], Consider the weighted voting system [15: 11, 7, 5, 2], What is the weight of the coalition {P1,P2,P4}. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. what are the non legislative powers of congress. Instead of looking at a player leaving a coalition, this method examines what happens when a player joins a coalition. stream Counting up times that each player is critical: Divide each players count by 16 to convert to fractions or percents: The Banzhaf power index measures a players ability to influence the outcome of the vote. /Filter /FlateDecode /Rect [188.925 2.086 190.918 4.078] As Im sure you can imagine, there are billions of possible winning coalitions, so the power index for the Electoral College has to be computed by a computer using approximation techniques. >> endobj /Contents 28 0 R This page titled 7.2: Weighted Voting is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Evaluate the source and summarize the article, then give your opinion of why you agree or disagree with the writers point of view. \hline \textbf { District } & \textbf { Times critical } & \textbf { Power index } \\ In other words: \[\frac{w_{1}+w_{2}+w_{3}+\cdots w_{N}}{2}> Calculate the Shapley-Shubik Power Index. \(\left\{P_{1}, P_{2}\right\}\) Total weight: 9. 23 0 obj << Underlining the critical players to make it easier to count: \(\left\{\underline{P}_{1}, \underline{P}_{2}\right\}\), \(\left\{\underline{P}_{1}, \underline{P}_{3}\right\}\). There are a lot of them! Suppose instead that the number of seats could be adjusted slightly, perhaps 10% up or down. endobj If in a head-to-head comparison a majority of people prefer B to A or C, which is the primary fairness criterion violated in this election? \(\mathrm{P}_{1}\) is pivotal 3 times, \(\mathrm{P}_{2}\) is pivotal 3 times, and \(\mathrm{P}_{3}\) is pivotal 0 times. Another sequential coalition is. As you can see, computing the Shapley-Shubik power index by hand would be very difficult for voting systems that are not very small. >> endobj There are 3! There are 4 such permutations: BAC, CAB, BCA, and CBA, and since 3! This coalition has a combined weight of 7+6+3 = 16, which meets quota, so this would be a winning coalition. Find a weighted voting system to represent this situation. 26 0 obj << \(7 !=7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=5040\). The downtown business association is electing a new chairperson, and decides to use approval voting. /Resources 23 0 R N
QB0)/%F['r/g}9AThuHo/$S9LoniA1=-a >> endobj Consider the voting system [16: 7, 6, 3, 3, 2]. In order for a motion to pass, it must have a minimum number of votes. A college offers tutoring in Math, English, Chemistry, and Biology. Consider the running totals as each player joins: P 3 Total weight: 3 Not winning P 3, P 2 Total weight: 3 + 4 = 7 Not winning P 3, P 2, P 4 Total weight: 3 + 4 + 2 = 9 Winning R 2, P 3, P 4, P 1 Total weight: 3 + 4 + 2 + 6 = 15 Winning How many sequential coalitions will there be in a voting system with 7 players? In the coalition {P1, P2, P4}, every player is critical. /A << /S /GoTo /D (Navigation48) >> We start by listing all winning coalitions. Describe how an alternative voting method could have avoided this issue. The total weight is . In the coalition {P1, P3, P4, P5}, any player except P1 could leave the coalition and it would still meet quota, so only P1 is critical in this coalition. In parliamentary governments, forming coalitions is an essential part of getting results, and a partys ability to help a coalition reach quota defines its influence. We will have 3! The votes are shown below. Additionally, they get 2 votes that are awarded to the majority winner in the state. /Border[0 0 0]/H/N/C[.5 .5 .5] \hline P_{1} & 3 & 3 / 6=50 \% \\ xXnF}WOrqEv -RX/EZ#H37n$bRg]xLDkUz/{e: }{qfDgJKwJ \!MR[aEO7/n5azX>z%KW/Gz-qy7zUQ7ft]zv{]/z@~qv4?q#pn%Z5[hOOxnSsAW6f --`G^0@CjqWCg,UI[-hW mnZt6KVVCgu\IBBdm%.C/#c~K1.7eqVxdiBtUWKj(wu9; 28FU@s@,x~8a Vtoxn` 9[C6X7K%_eF1^|u0^7\$KkCgAcm}kZU$zP[G)AtE4S(fZF@nYA/K]2Y>>| K
2K`)Sd90%Yfe:K;oi. Survival Times | A player will be a dictator if their weight is equal to or greater than the quota. Since the quota is 16, and 16 is more than 15, this system is not valid. A coalition is any group of players voting the same way. This means player 5 is a dummy, as we noted earlier. \hline \textbf { District } & \textbf { Weight } \\ Counting Problems To calculate these power indices is a counting problem. Their results are tallied below. This will put the ! Previously, the coalition \(\left\{P_{1}, P_{2}\right\}\) and \(\left\{P_{2}, P_{1}\right\}\) would be considered equivalent, since they contain the same players. P_{3}=1 / 5=20 \% 19 0 obj << /Resources 12 0 R Math 100: Liberal Arts Mathematics (Saburo Matsumoto), { "8.01:_Apportionment" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.02:_Apportionment_of_Legislative_Districts" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.03:_Voting_Theory" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.04:_Weighted_Voting" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Mathematics_and_Problem-Solving" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Mathematics_and_Numbers" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Mathematics_and_Finance" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Probability_and_Odds" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Data_and_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Growth_and_Decay" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Mathematics_and_the_Arts" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Mathematics_and_Politics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Selected_Topics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Appendix" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "factorial", "license:ccby", "Banzhaf power index", "Shapley-Shubik power index", "weighted voting" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FCollege_of_the_Canyons%2FMath_100%253A_Liberal_Arts_Mathematics_(Saburo_Matsumoto)%2F08%253A_Mathematics_and_Politics%2F8.04%253A_Weighted_Voting, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Calculating Power: Shapley-Shubik Power Index, status page at https://status.libretexts.org, In each coalition, identify the players who are critical, Count up how many times each player is critical, Convert these counts to fractions or decimals by dividing by the total times any player is critical, In each sequential coalition, determine the pivotal player, Count up how many times each player is pivotal, Convert these counts to fractions or decimals by dividing by the total number of sequential coalitions. Meets quota. In the weighted voting system \([17: 12,7,3]\), determine which player(s) are critical player(s). The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. \(\left\{P_{1}, P_{2}\right\}\) Total weight: 9. No player is a dictator, so well only consider two and three player coalitions. Find the Shapley-Shubik power index for the weighted voting system \(\bf{[36: 20, 17, 15]}\). /D [9 0 R /XYZ 334.488 0 null] Then, when player two joins, the coalition now has enough votes to win (12 + 7 = 19 votes). /D [9 0 R /XYZ 334.488 0 null] \end{array}\). The individual ballots are shown below. First, note that , which is easy to do without the special button on the calculator, be we will use it anyway. 25 0 obj << /ProcSet [ /PDF /Text ] /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R the voter whose immediate sequential presence changes the vote from lose to win. This is the same answer as the Banzhaf power index. If the legislature has 116 seats, apportion the seats using Hamiltons method. Find an article or paper providing an argument for or against the Electoral College. Translated into a weighted voting system, assuming a simple majority is needed for a proposal to pass: Listing the winning coalitions and marking critical players: \(\begin{array} {lll} {\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}\}} & {\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}, \mathrm{NH}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{NH}, \mathrm{LB}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}\}} & {\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}, \mathrm{LB}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{NH}, \mathrm{GC}}\} \\{\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}\}} & {\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}, \mathrm{GC}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{LB}, \mathrm{GC}}\} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{NH}\}} & {\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}, \mathrm{NH}, \mathrm{LB}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{NH}, \mathrm{LB}, \mathrm{GC}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{LB}\}} & {\{\underline{\mathrm{H} 1}, \mathrm{OB}, \mathrm{NH}, \mathrm{GC}\}} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{GC}\}} & {\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}, \mathrm{LB}, \mathrm{GC}\}} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}, \mathrm{NH}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{NH}, \mathrm{LB}\}} & {\{\underline{\mathrm{H} 1}, \underline{\mathrm{OB}}, \mathrm{NH}, \mathrm{LB} . Consider the voting system \([16: 7, 6, 3, 3, 2]\). xVMs0+t$c:MpKsP@`cc&rK^v{bdA2`#xF"%hD$rHm|WT%^+jGqTHSo!=HuLvx TG9;*IOwQv64J) u(dpv!#*x,dNR3 4)f2-0Q2EU^M: JSR0Ji5d[ 1 LY5`EY`+3Tfr0c#0Z\! Legal. In weighted voting, we are most often interested in the power each voter has in influencing the outcome. if n is the number of players in a weighted voting system, then the number of coalitions is this. A company has 5 shareholders. They are trying to decide whether to open a new location. The winner is then compared to the next choice on the agenda, and this continues until all choices have been compared against the winner of the previous comparison. Shapley-Shubik Power (Chapter 2 Continued) Sequential coalitions - Factorial - Pivotal Player - Pivotal count - Shapley-Shubik Power Index (SSPI) - Ex 6 (LC): Given the following weighted voting system: [10: 5, 4, 3, 2, 1] a) How many Sequential Coalitions will there be? Next we determine which players are critical in each winning coalition. It looks like if you have N players, then you can find the number of sequential coalitions by multiplying . Under the same logic, players one and two also have veto power. Chi-Squared Test | Reapportion the previous problem if the store has 25 salespeople. We start by listing all winning coalitions. When a person goes to the polls and casts a vote for President, he or she is actually electing who will go to the Electoral College and represent that state by casting the actual vote for President. K\4^q@4rC]-OQAjp_&.m5|Yh&U8 @u~{AsGx!7pmEy1p[dzXJB$_U$NWN_ak:lBpO(tq@!+@S ?_r5zN\qb >p Ua 13 0 obj << /MediaBox [0 0 362.835 272.126] Sequence Calculator Step 1: Enter the terms of the sequence below. /Filter /FlateDecode The sequential coalitions for three players (P1, P2, P3) are: . \(< P_{1}, \underline{P}_{2}, P_{3} > \quad < P_{1}, \underline{P}_{3}, P_{2} > \quad< P_{2}, \underline{P}_{1_{2}} P_{3} >\), \( \quad \quad \). \hline \text { Hempstead #2 } & 31 \\ 14 0 obj << Describe how Plurality, Instant Runoff Voting, Borda Count, and Copelands Method could be extended to produce a ranked list of candidates. This calculation is called a factorial, and is notated \(N!\) The number of sequential coalitions with \(N\) players is \(N!\). /Rect [188.925 2.086 190.918 4.078] endobj /Length 756 Does this illustrate any apportionment issues? Combining these possibilities, the total number of coalitions would be:\[N(N-1)(N-2)(3-N) \ldots(3)(2)(1)\nonumber \]This calculation is called a factorial, and is notated \(N !\) The number of sequential coalitions with \(N\) players is \(N !\). \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{4}\right\} \quad \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{5}\right\}\\ Since the quota is 9, and 9 is more than 8.5 and less than 17, this system is valid. stream Since player 1 and 2 can reach quota with either player 3 or player 4s support, neither player 3 or player 4 have veto power. Legal. Then player three joins but the coalition is still a losing coalition with only 15 votes. \hline \text { Glen Cove } & 0 & 0 / 48=0 \% \\ Find the Banzhaf power index for the voting system \([8: 6, 3, 2]\). Calculate the winner under these conditions. Well begin with some basic vocabulary for weighted voting systems. Notice that player 5 has a power index of 0, indicating that there is no coalition in which they would be critical power and could influence the outcome. The two methods will not usually produce the same exact answer, but their answers will be close to the same value. After hiring that many new counselors, the district recalculates the reapportion using Hamilton's method. Meets quota. The total weight is . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Some people feel that Ross Perot in 1992 and Ralph Nader in 2000 changed what the outcome of the election would have been if they had not run. In the weighted voting system \([57: 23,21,16,12]\), are any of the players a dictator or a dummy or do any have veto power. /Font << /F43 15 0 R /F20 17 0 R /F16 16 0 R /F22 26 0 R /F32 27 0 R /F40 28 0 R /F21 29 0 R >> To calculate the Shapley-Shubik Power Index: How many sequential coalitions should we expect to have? Now we count up how many times each player is pivotal, and then divide by the number of sequential coalitions. Since the quota is 9, and 9 is between 7.5 and 15, this system is valid. A pivotal player is the player in a sequential coalition that changes a coalition from a losing coalition to a winning one. If the legislature has 200 seats, apportion the seats. /Resources 1 0 R Also, no two-player coalition can win either. /Length 1197 Sample Size Calculator | /Type /Page Lets examine these for some concepts. To decide on a new website design, the designer asks people to rank three designs that have been created (labeled A, B, and C). Consider the weighted voting system [31: 10,10,8,7,6,4,1,1], Consider the weighted voting system [q: 7,5,3,1,1]. Once you choose one for the first spot, then there are only 2 players to choose from for the second spot. ; U_K#_\W )d > . Calculate the percent. \hline \textbf { Player } & \textbf { Times pivotal } & \textbf { Power index } \\ Each state has a certain number of Electoral College votes, which is determined by the number of Senators and number of Representatives in Congress. >> Notice that player 1 is not a dictator, since player 1 would still need player 2 or 3s support to reach quota. /Filter /FlateDecode \hline \text { Oyster Bay } & 28 \\ In the Electoral College, states are given a number of votes equal to the number of their congressional representatives (house + senate). The Banzhaf power index is one measure of the power of the players in a weighted voting system. Will not usually produce the same logic, players one and two have. 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A college offers tutoring in Math, English, Chemistry, and decides to approval... 188.925 2.086 190.918 4.078 ] endobj /Length 756 Does this illustrate any apportionment issues determine. Some concepts it anyway that many new counselors, the District recalculates the reapportion using Hamilton 's.. Many new counselors, the District recalculates the reapportion using Hamilton 's method are most often interested the. | a player leaving a coalition, this method examines what happens when player.