Explanation Given Information: Banzhaf Power indices Concept used: B P I ( v ) = Number of times voter v is a critical voter Nuner of times any voter is a critical voter Calculation: Winning coalition Number of votes Critical Voters { A , B } 7 A , B { A , C } 6 A , C { A , B , C } 9 A { A , B , D } 8 A , B { A , C , D } 7 A , C { B , C , D } 6 B , C , D { A , B , C , D } 10 None Using the Banzhaf power index, we have B P I ( A ) = 5 / 12 . Consider A for Belgium, B for France, C for Germany, D for Italy, E for Luxembourg, F for Netherlands, { 12 : 2 , 4 , 4 , 4 , 1 , 2 } . Winning coalition Number of votes Critical Voters { B , C , D } 12 B , C , D { A , B , C , D } 14 B , C , D { A , B , C , F } 12 A , B , C , F { B , C , D , E } 13 B , C , D { B , C , D , F } 14 B , C , D { A , B , C , D , E } 15 B , C , D { A , B , C , D , F } 17 None { B , C , D , E , F } 15 B , C , D { A , B , C , D , E , F } 17 None Now draw the blocking Coalition table. Winning coalition Number of votes Number of remaining votes Critical Voters { A , B , C } 10 7 A , B , C { A , B , D } 10 7 A , B , D { B , C , D } 12 5 B , C , D { B , C , E } 9 8 B , C , E { B , C , F } 10 7

Mathematical Excursions (MindTap Course List)

4th Edition

ISBN: 9781305965584

Author: Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Clegg

Publisher: Cengage Learning

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Chapter 4.3, Problem 4EE

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Calculate the Banzhaf power indices for Belgium and Luxembourg using the alternative definition.

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Chapter 4 Solutions

Mathematical Excursions (MindTap Course List)

Ch. 4.1 - Verify this apportionment using the Jefferson...Ch. 4.1 - Find the apportionment that would have resulted if...Ch. 4.1 - Prob. 3EE Ch. 4.1 - Find the apportionment that would have resulted if...Ch. 4.1 - Explain how to calculate the standard divisor of...Ch. 4.1 - Teacher Aides A total of 25 teacher aides are to...Ch. 4.1 - In the Hamilton apportionment method, explain how...Ch. 4.1 - Prob. 4ES Ch. 4.1 - Governing Boards The following table shows how the...Ch. 4.1 - Forest Rangers The table below shows how the...

Ch. 4.1 - Sales Associates The table below shows the number...Ch. 4.1 - Hospital Interns The table below shows the number...Ch. 4.1 - House of Representatives The U.S. House of...Ch. 4.1 - College Enrollment The following table shows the...Ch. 4.1 - Medical Care A hospital district consists of six...Ch. 4.1 - What is the Alabama paradox?Ch. 4.1 - What is the population paradox?Ch. 4.1 - What is the new states paradox?Ch. 4.1 - 15. What is the Balinski-Young Impossibility...Ch. 4.1 - Apportionment of Projectors Consider the...Ch. 4.1 - Hotel Management A company operates four resorts....Ch. 4.1 - Prob. 18ES Ch. 4.1 - Management Scientific Research Corporation has...Ch. 4.1 - Prob. 20ES Ch. 4.1 - Elementary School Teachers The following table...Ch. 4.1 - Social Workers The following table shows the...Ch. 4.1 - Computer Usage The table below shows the number of...Ch. 4.1 - The population of Illinois increased by over...Ch. 4.1 - Prob. 25ES Ch. 4.1 - Prob. 26ES Ch. 4.1 - Computer Usage Use the Webster method to apportion...Ch. 4.1 - Demographics The table below shows the populations...Ch. 4.1 - Which of she following apportionment methods can...Ch. 4.1 - According to Michael Balinski and H. Peyton Young,...Ch. 4.1 - 31. What method is presently used to apportion the...Ch. 4.1 - Prob. 32ES Ch. 4.1 - John Quincy Adams, the sixth president of the...Ch. 4.1 - In the Huntington-Hill method of apportionment,...Ch. 4.1 - Prob. 35ES Ch. 4.2 - Using the Borda method of voting, which flavor of...Ch. 4.2 - Instead of using the normal Borda method, suppose...Ch. 4.2 - Prob. 3EE Ch. 4.2 - Suppose the Borda method used in Exercise I of...Ch. 4.2 - Prob. 5EE Ch. 4.2 - Can the assignment of points for first. Second,...Ch. 4.2 - Prob. 7EE Ch. 4.2 - Prob. 1ES Ch. 4.2 - Explain why the plurality voting system may not be...Ch. 4.2 - Prob. 3ES Ch. 4.2 - Explain how the plurality with elimination voting...Ch. 4.2 - Prob. 5ES Ch. 4.2 - Prob. 6ES Ch. 4.2 - Is there a best voting method? Is one method more...Ch. 4.2 - Explain why. if only two candidates are running,...Ch. 4.2 - Presidential Election The table below shows the...Ch. 4.2 - Breakfast Cereal Sixteen people were asked to rank...Ch. 4.2 - Cartoon Characters A kindergarten class was...Ch. 4.2 - Catering A 15-person committee is having lunch...Ch. 4.2 - Movies Fifty consumers were surveyed about their...Ch. 4.2 - Breakfast Cereal Use the Borda count method of...Ch. 4.2 - Prob. 15ES Ch. 4.2 - Catering Use the Borda count method of voting to...Ch. 4.2 - Class Election A senior high school class held an...Ch. 4.2 - Cell Phone Usage A journalist reviewing various...Ch. 4.2 - Baseball Uniforms A Little League baseball team...Ch. 4.2 - Radio Stations A number of college students were...Ch. 4.2 - Class Election Use plurality with elimination to...Ch. 4.2 - Prob. 22ES Ch. 4.2 - Campus Club A campus club has money left over in...Ch. 4.2 - Recreation A company is planning its annual summer...Ch. 4.2 - X-Men Movies Fans of the X-Men movies have been...Ch. 4.2 - Prob. 26ES Ch. 4.2 - School Mascot A new college needs to pick a moscot...Ch. 4.2 - Election Five candidates are running for president...Ch. 4.2 - Prob. 29ES Ch. 4.2 - Radio Stations Use the pairwise comparison method...Ch. 4.2 - Does the winner in Exercise tic satisfy the...Ch. 4.2 - 32. Does the winner in Exercise 12 satisfy the...Ch. 4.2 - Prob. 33ES Ch. 4.2 - Prob. 34ES Ch. 4.2 - Prob. 35ES Ch. 4.2 - Prob. 36ES Ch. 4.2 - Election Three candidates are running for mayor. A...Ch. 4.2 - Film Competition Three films have been selected as...Ch. 4.2 - 39. Election A campus club needs to elect four...Ch. 4.2 - Scholarship Awards The members of a scholarship...Ch. 4.2 - Another method of voting is to assign a weight. or...Ch. 4.2 - Prob. 42ES Ch. 4.2 - Prob. 43ES Ch. 4.3 - Using the data in Example 1 on page 211, list all...Ch. 4.3 - For the data in Example I on page 211, calculate...Ch. 4.3 - Prob. 3EE Ch. 4.3 - Prob. 4EE Ch. 4.3 - Create a voting system with three members that is...Ch. 4.3 - Prob. 6EE Ch. 4.3 - Prob. 7EE Ch. 4.3 - In the following exercises that involve weighted...Ch. 4.3 - In the following exercises that involve weighted...Ch. 4.3 - Prob. 3ES Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Prob. 8ES Ch. 4.3 - Prob. 9ES Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Calculate, if possible, the l3anzhaf power index...Ch. 4.3 - Prob. 13ES Ch. 4.3 - Prob. 14ES Ch. 4.3 - Music Education A music department Consists of a...Ch. 4.3 - Prob. 16ES Ch. 4.3 - Criminal Justice In a criminal trial, each of the...Ch. 4.3 - Criminal Justice In California civil court cases,...Ch. 4.3 - Identify any dictator and all dummies for each...Ch. 4.3 - Identify any dictator and all dummies for each...Ch. 4.3 - Prob. 21ES Ch. 4.3 - Prob. 22ES Ch. 4.3 - Football At the beginning of each football season,...Ch. 4.3 - Prob. 24ES Ch. 4.3 - Prob. 25ES Ch. 4.3 - Prob. 26ES Ch. 4.3 - Consider the weighted voting system { q:8,3,3,2 },...Ch. 4.3 - Prob. 28ES Ch. 4.3 - Prob. 29ES Ch. 4.3 - UN Security Council The United Nations C Security...Ch. 4 - Education The following table shows the...Ch. 4 - Airline Industry The following table shows the...Ch. 4 - Airline Industry The table below shows how the...Ch. 4 - 4. Education The following table shows the number...Ch. 4 - Technology A company has four offices. The...Ch. 4 - Automobile Sales Consider the apportionment of 27...Ch. 4 - Music Company MusicGalore.biz has offices in Los...Ch. 4 - Building Inspectors A city apportions 34 building...Ch. 4 - Prob. 9RE Ch. 4 - Prob. 10RE Ch. 4 - Corporate Security The Huntington-Hill...Ch. 4 - Essay Contest Four finalists are competing in an...Ch. 4 - Ski Club A campus ski club is trying o decide...Ch. 4 - Prob. 14RE Ch. 4 - Consumer Preferences A group of consumers were...Ch. 4 - Prob. 16RE Ch. 4 - Prob. 17RE Ch. 4 - Homecoming Queen Three high school students are...Ch. 4 - Prob. 19RE Ch. 4 - Prob. 20RE Ch. 4 - Prob. 21RE Ch. 4 - A weighted voting system for voters A. B. C. D....Ch. 4 - Prob. 23RE Ch. 4 - Prob. 24RE Ch. 4 - 25. Calculate the Banzhaf power indices for voters...Ch. 4 - Calculate the Banzhaf power indices for voters A,...Ch. 4 - Prob. 27RE Ch. 4 - Identify any dictator and all dummies for each...Ch. 4 - Four voters. A. B. C. and D. make decisions by...Ch. 4 - Postal Service The table below shows the number of...Ch. 4 - Computer Allocation The following table shows the...Ch. 4 - High School Counselors The following table shows...Ch. 4 - Prob. 4T Ch. 4 - Consumer Preference One hundred consumers ranked...Ch. 4 - Prob. 6T Ch. 4 - Exam Review A professor is preparing an extra...Ch. 4 - Prob. 8T Ch. 4 - Drama Department The four staff members. A, B,C,...Ch. 4 - Three voters. A. B. and C, make decisions by using...

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