Voting and Apportionment(Due with Final Exam)

Voting and Apportionment(Due with Final Exam) The XYZ Takeaway W Affair. 1. Consider the following preference table for candidates x, y, z, and w. Number of votes 200 150 250 300 100 First choice z y x w y Second choice x z y z x Third choice y x z x w Fourth choice w w w y z a) Is there a Pairwise comparisons winner? If so, which candidate? b) Use the Borda Count method to determine a winner. c) Delete candidate w, and redo the Borda Count method to determine a winner. d) Does the Pairwise comparisons winner change after deleting candidate w? Gimme An A, Gimme A C. 2. Consider the following preference table for candidates A, B, and C. a) Who is the Plurality winner? b) Who wins the Pairwise comparisons? c) Who is the Borda Count winner? Number of votes 27 24 2 First choice A B C Second choice C C B Third choice B A A d) If candidate B drops out, does the Borda Count winner change?

Gimme A B, Gimme A C. 3. Consider the following preference table for candidates A, B, and C. Number of votes 20 19 5 First choice A B C Second choice B C B Third choice C A A a) Who wins with Plurality with elimination? b) If candidate A drops out, will the Plurality with elimination winner change? In 1822, Congress discussed, but never adopted, a method similar to Hamilton s Method proposed by the South Carolina Congressman William Lowndes shortly before he died at sea. With Lowndes s Method, the initial allocation of seats is made just as with Hamilton s Method by giving each state its lower quota. Any remaining seats are assigned to the states with the largest relative fractional parts, i. e. the fractional part divided by the lower quota. For instance, if a state has a standard quota of 14.37, then the relative fractional part is.37.0264 14. I ve Got A Sinking Feeling About This Apportionment. 4. Use Lowndes Method to apportion 28 seats among the following 5 counties: County Cheshire Grafton Hillsborough Rockingham Strafford Total Population 28,772 13,472 32,871 43,169 23,601 141,885 The current method of apportionment used by the House of Representatives is the Huntington- Hill Method. The method is similar to Jefferson, Adams, and Webster in that a modified divisor is found, but instead of rounding down, up, or to the nearest integer, you compare the modified quota(population divided by modified divisor) to the geometric mean of the modified upper and lower quotas. If the modified quota for a particular state is 15.542, then the geometric mean of the modified upper and lower quotas is 1516 15.4919. Since 15.542 15.4919..., this particular state would be allotted 16 seats.

Over Huntington-Hill, Over Dale. 5. The Dale County student council will consist of 26 representatives from the five schools in Dale County. Determine the Huntington-Hill apportionment for each of the five schools using a modified divisor of 960: School Student Population Modified Quota North High South High Valley High Meadow High Ridge High Total 9,061 7,179 5,259 3,319 1,182 26,000 9,061 9.439 960 Geometric Mean Apportionment 9 910 9.487 Mirror, Mirror, On The Wall, Which Is The Fairest Of Them All? 6. Compare the Huntington-Hill apportionment from the previous problem to the other apportionments, and comment on which one seems fairest. North High South High Valley High Meadow High Ridge High Hamilton 9 7 5 4 1 Jefferson 10 7 5 3 1 Adams 9 7 5 3 2 Webster 9 8 5 3 1

Huntington-Hill For Mountain County. 7. Mountain County is forming an all-county handball team consisting of 13 players. The handball committee has decided to use the Huntington-Hill apportionment in deciding the number of players to represent each of the 6 clubs in the county. a) Determine the apportionment with a modified divisor of 100: Club Rainier Adams Whitney St. Helens Hood Shasta Total # of players 501 332 53 46 15 58 1005 Modified Quota 5.01 Geometric Mean 5.477 # of handball 5 players Actual Quota 6.48 b) Does the apportionment satisfy quota? Hunting For An Apportionment. 8. Use the Huntington-Hill method to apportion SuperTec s board for a board of trustees with six members and a modified divisor of 265: Department Development Production Sales Service Total Student Population 241 673 368 173 1455 Modified Quota.9094 Geometric Mean 0 # of trustees 1

Still Hunting For An Apportionment. 9. The apportionment for the board from the previous problem using the other methods are given below: Development Production Sales Service Hamilton 2 2 1 1 Jefferson 1 3 1 1 Adams 1 2 2 1 Webster 1 3 1 1 Which method would you recommend? Explain. Hunting For Another Seat. 10. According to the 2000 census, the population of California was 33,930,798 and the population of Utah was 2,236,714. California was apportioned 53 House seats, and Utah received 3 House seats. If the governor of Utah believes that his state was undercounted, what increase in population would be large enough to possibly entitle Utah to take a seat from California if the apportionment is by Huntington-Hill? Hint: Let s start by taking a seat away from California. This leaves 52 seats for California. In order for it to be possible for Utah to get California s seat, you d have to find a Population of California new modified divisor, d, so that 52 53 and d Revised population of Utah 34. The first inequality rearranges into d Population of California Revised population of Utah d, and the second into d. 52 53 34 If you put the two inequalities together, you get Revised population of Utah Population of California. Plugging in the population 34 52 53 Revised population of Utah 33,930,798 of California leads to. Solve the 34 52 53 inequality, and use it to answer the question.

Threesome Or Twosome. 11. Here are the estimated results of the 1980 U. S. Senate race in New York among Alphonse D Amato(D), Elizabeth Holtzman(H), and Jacob Javits(J): Percentage of votes 22 23 15 29 7 4 First choice D D H H J J Second choice H J D J H D Third choice J H J D D H a) Determine the Plurality winner. b) Determine the Borda Count winner. c) Determine the Plurality-with-elimination winner. d) Determine the Pairwise Comparison winner. e) If J dropped out of the race, determine the winner using the previous 4 methods, and comment on the Irrelevant Alternatives Criterion. Spock And Martin-A Pair A Docs. 12. A country has four states, A, B, C, and D. Its house of representatives has 100 members, apportioned by the Hamilton Method. A new census is taken, and the house is reapportioned. Here s the information: State Old census New census A 5,525,381 5,657,564 B 3,470,152 3,507,464 C 3,864,226 3,885,693 D 201,203 201,049 Total 13,060,962 13,251,770 a) Apportion the house using the old census. b) Reapportion using the new census. c) Has the population paradox occurred? Explain.

Not So Sweet Home Alabama Paradox. 13. A country has five states with populations of 5,576,330, 1,387,342, 3,334,241, 7,512,860, and 310,968. Its house of representatives is apportioned by the Hamilton Method. a) Calculate the apportionments for house sizes of 82, 83, and 84. Does the Alabama paradox occur? b) Calculate the apportionments for house sizes of 89, 90, and 91. Does the Alabama paradox occur? Smorgasborda. 14. In an election, there are 3 candidates and 25 voters. a) What is the maximum number of points that a candidate can receive using the Borda count method? b) What is the minimum number of points that a candidate can receive using the Borda count method? Smorgasborda Ala Carte. 15. In an election, there are 4 candidates and 75 voters. a) What is the maximum number of points that a candidate can receive using the Borda count method? b) What is the minimum number of points that a candidate can receive using the Borda count method?

Highs And Lows By Pairs. 16. In an election, there are 7 candidates. a) What is the maximum number of points that a candidate can receive using the Pairwise comparison method? b) What is the minimum number of points that a candidate can receive using the Pairwise comparison method? More Highs And Lows By Pairs. 17. In an election, there are 9 candidates. a) What is the maximum number of points that a candidate can receive using the Pairwise comparison method? b) What is the minimum number of points that a candidate can receive using the Pairwise comparison method? I Know What You Did Last Summer. 18. The Jefferson High School Backpackers Club schedules a 5-day excursion each summer. The 10 members have narrowed their choices for the upcoming summer to the national parks at Denali(D), Yosemite(Y), and Zion(Z), and they decide to vote to determine the final location. The individual ballots are shown below. First D Y Y Z D Z Z Y D D Second Y Z Z D Y Y Y Z Z Y Third Z D D Y Z D D D Y Z a) Write the preference table for this election. b) Did any of the locations receive a majority of first-place votes in this election? If so, which one? c) Did any of the locations receive a plurality of the first-place votes in this election? If so, which one? d) By assigning 3 points for first place, 2 points for second place, and 1 point for third place, determine the winner using the Borda count method. Is this different from the plurality winner?

Clinton Said To His Opponents: Close, But No Cigar. His Opponents Said: Thank Goodness! 19. In the 1992 U. S. presidential election, Bill Clinton (C) received 43% of the popular vote; George H. W. Bush(B) 37%; and H. Ross Perot(P) 19%. Suppose that each voter had completed a preference ballot instead of just voting for one candidate and that the resulting preference table is below. 36% 7% 31% 6% 14% 5% First C C B B P P Second P B P C B C Third B P C P C B Determine the winner using the Borda count method. The Hare system, developed by Thomas Hare in 1861, uses preference ballots, but they are tabulated according to a different scheme. In this system, the candidate with the fewest firstplace votes is eliminated on every ballot, and then the preference table is adjusted by advancing each candidate ranked under the deleted one. The process continues until one candidate receives a majority of votes. Here s an example: In this election, there are 4 candidates: Albert(A), Barbara(B), Chris(C), and Derek(D), and here s the preference table: 27 23 22 13 11 4 First B D A C C B Second C A C B D A Third D B D A A C Fourth A C B D B D In the first round, the number of first-place votes for each candidate are 31 for Barbara, 24 for Chris, 23 for Derek, and 22 for Albert. Since no one has a majority at this point, Albert is eliminated. 27 23 22 13 11 4 First B D C C C B Second C B D B D C Third D C B D B D

In the second round, the number of first-place votes for each remaining candidate are 31 for Barbara, 46 for Chris, and 23 for Derek. Since no one has a majority at this point, Derek is eliminated. 27 23 22 13 11 4 First B B C C C B Second C C B B B C In the third round, Barbara has 54 first-place votes, which is a majority, so she wins the election. Lost It By A Hare. 20. A football team must choose from offers to a bowl game in Miami(M), Phoenix(P), and San Diego(S). The preference table is shown below. 19 16 12 7 First M S P S Second S P M M Third P M S P Determine the winner using Plurality, Hare, and Borda methods. Are the winners different? If You Aint Got Hare, You Don t Need Coombs. 21. The Coombs method proceeds in a similar manner to the Hare system except that the candidate eliminated in each round is the one with the most last place votes. Consider the following preference table. 7 3 3 2 First A B C C Second B C B A Third C A A B a) Determine the winner using the Hare system. b) Determine the winner using the Coombs method.

Weighted Voting Systems There are many types of voting systems. When people are asked to vote for or against a resolution, a one-person, one-vote majority system is often used to decide the outcome. In this type of voting, each voter receives one vote, and the resolution passes only if it receives most of the votes. In any voting system, the number of votes that is required to pass a resolution is called the quota. A coalition is a set of voters each of whom votes the same way, either for or against a resolution. A winning coalition is a set of voters the sum of whose votes is greater than or equal to the quota. A losing coalition is a set of voters the sum of whose votes is less than the quota. Sometimes you can find all the wining coalitions in a voting process by making an organized list. For instance, consider the committee consisting of Alice, Barry, Cheryl, and Dylan. To decide on any issues, they use a one-person, one-vote majority system. Since each of the four voters has a single vote, the quota for this majority voting system is 3. The winning coalitions consist of all subsets of the voters that have 3 or more people. These winning coalitions are listed in the following table with A representing Alice, B representing Barry, C representing Cheryl, and D representing Dylan. Winning Coalition Sum of the votes A, B, C 3 A, B, D 3 A, C, D 3 B, C, D 3 A, B, C, D 4 A weighted voting system is one in which some voters votes carry more weight regarding the outcome of an election. As an example, consider the selection committee that consists of four people designated by A, B, C, and D. Voter A s vote has a weight of 2, and the vote of each other member of the committee has a weight of 1. The quota for this weighted system is 3. A winning coalition must have a weighted voting sum of at least 3. The winning coalitions are listed in the following table. Winning Coalition Sum of the weighted votes AB, 3 AC, 3 AD, 3 B, C, D 3 A, B, C 4 A, B, D 4 A, C, D 4 A, B, C, D 5

A minimal winning coalition is a winning coalition that has no proper subset that is a winning coalition. In a minimal winning coalition each voter is said to be a critical voter, because if any of the voters leaves the coalition, the coalition will then become a losing coalition. In the previous table, the minimal winning coalitions are AB,, AC,, AD,, and B, C, D. If any single voter leaves one of these coalitions, then the coalition will become a losing coalition. The coalition A, B, C, D is not a minimal winning coalition, because it contains at least one proper subset, for instance AB,, that is a winning coalition. A weighted voting system of n voters can be written as q : w1, w2,, w n, where q is the quota and w 1 through w n represent the weights of each of the n voters. We All Weigh The Same. 22. A selection committee consists of Ryan, Susan, and Trevor. To decide on issues they use a one-person, one-vote majority system. 2:1,1,1 a) Find all the winning coalitions. b) Find all losing coalitions. Some Weigh More Than Others. 23. A selection committee consists of three people designated by M, N, and P. M s vote has a weight of 3, N s vote a weight of 2, and P s vote has a weight of 1. The quota for this weighted voting system is 4. 4:3,2,1 a) Find all winning coalitions. b) Find all minimal winning coalitions.

Banzhaf Power Index There are a number of different measures of the power of a voter in a weighted voting system. One measure called the Banzhaf power index was derived by John Banzhaf in 1965. The Banzhaf power index of a voter v, symbolized by BPI(v), is given by number of times voter v is a critical voter BPI v. As an example, consider the four number of times any voter is a critical voter people A, B, C, and D and the one-person, one-vote system given by {3:1,1,1,1}. Winning coalition Number of votes Critical voters A, B, C 3 A,B,C A, B, D 3 A,B,D A, C, D 3 A,C,D B, C, D 3 B,C,D A, B, C, D 4 None To find BPI(A), we look under the critical voters column and find that A is a critical voter 3 times. The number of times that any voter is a critical voter is 12. So we conclude that 3 1 1 1 BPI A. Similarly, we can calculate that BPI B, BPI C, and 12 4 4 4 1 BPI D. In this case, each voter has the same power. This is expected in a voting system 4 where each voter has one vote. Now suppose that three people A, B, and C have the voting system {3:3,1,1}. Winning coalition Number of votes Critical voters A 3 A,,,, AB 4 A AC 4 A A B C 5 A 4 1 4 0 0 4 In this case, BPI A, BPI B, and 0 BPI C 0. Voter A has all the power. 4

A Different Kind Of Stock Index. 24. Suppose that the stock in a company is held by 5 people: A, B, C, D, and E. The voting system for this company is {626:350, 300, 250,200,150}. Winning coalition Critical voters Winning coalition Critical voters AB, A,B B, C, E B,C,E A, B, C A,B B, D, E B,D,E,, A, B, C, D None A B D A,B A, B, E A,B A, B, C, E A, C, D A,C,D A, B, D, E A, C, E A,C,E A, C, D, E A, D, E A,D,E B, C, D, E B, C, D B,C,D A, B, C, D, E a) Find the Banzhaf power index for each of the five voters. b) Which voter(s) has(have) the most power? c) Which voter(s) has(have) the least power? None None A B None Don t Tie Me Down. 25. Suppose that a city council consists of four members A, B, C, and D, and a mayor M, each with an equal vote. The mayor votes only when there is a tie among the members of the council. In all cases, a resolution passes if it receives 3 or more votes. Winning coalition, without the mayor,, Critical voters Winning coalition, with the mayor A, B, M A B C A,B,C A, B, D A,B,D A, C, M A, C, D A,C,D A, D, M B, C, D B,C,D B, C, M A, B, C, D None B, D, M C, D, M Critical voters A,B,M A,C,M A,D,M B,C,M B,D,M C,D,M a) Find the Banzhaf power index for each of the five voters. b) Does the mayor have a different voting power than the council members?

Dictator Or Dummy? 26. A voter who has a weight greater than or equal to the quota is called a dictator. In a weighted voting system, the dictator has all the power. A voter who is never a critical voter has no power and is referred to as a dummy. This term is not meant to be a comment on the voter s intellectual ability, rather it just indicates that the voter has no ability to influence an election. Identify any dictator and all dummies for the following weighted voting systems: a) {16:16,5,4,2,1} b) {15:7,5,3,2} c) {19:12,6,3,1} d) {45:40,6,2,1} Is Borda Your Cup Of Tea? 27. Sixty people were asked to select their preferences among plain iced tea, lemon-flavored iced tea, and raspberry flavored iced tea. The preference table is below. Number of votes 25 20 15 1 st P R L 2 nd L L R 3 rd R P P a) Using the Borda Count Method, which flavor is preferred? Which is second? Which is third? b) If you used 1 point for first, 0 points for second, and -1 points for third, does this alter the preferences in part a)? c) If you used 10 points for first, 5 points for second, and 0 points for third, does this alter the preferences in part a)? d) If you used 20 points for first, 5 points for second, and 0 points for third, does this alter the preferences in part a)? e) If you used 25 points for first, 5 points for second, and 0 points for third, does this alter the preferences in part a)? f) Can the assignment of points for first, second, and third change the preference order when the Borda Count Method is used? g) Suppose the assignment of points for first, second, and third place are consecutive integers. Can the value of the starting integer change the outcome of the preferences?

Another method of voting is to assign a weight or score to each candidate rather than ranking the candidates in order. All candidates must receive a score, and two or more candidates can receive the same score from a voter. A score of 5 represents the strongest endorsement of a candidate. The scores range down to 1, which corresponds to complete disapproval of a candidate. A score of 3 represents indifference. The candidate with the most total points wins the election. Throwing Your Weight Around As A Voter. 28. The results of an election with weights is given in the following table. a) Find the winner of the election. # of votes 26 42 19 33 24 8 24 33 A 2 1 2 5 4 2 4 5 B 5 3 5 3 2 5 3 2 C 4 5 4 1 4 3 2 1 D 1 3 2 4 5 1 3 2 b) If plurality were used(assuming that a person s vote would go to the candidate that the person gave the highest score to), find the winner of the election.

Approval voting is a system in which voters may vote for more than one candidate. Each vote counts equally, and the candidate with the most total votes wins the election. Many people feel that this is a better system for large elections than simple plurality because it considers a voter s second choices, and is a stronger measure of overall voter support for each candidate. Some organizations use approval voting to elect their officers. The United Nations uses this method to elect the secretary-general. Math Goes To The Movies, But Only If You Approve. 29. Suppose a math class is going to show a film involving mathematics or mathematicians on the last day of class. The options are Stand and Deliver, Good Will Hunting, A Beautiful Mind, Proof, and Contact. The students vote using approval voting. The results are as follows. 8 students vote for all five films. 8 students vote for Good Will Hunting, A Beautiful Mind, and Contact. 8 students vote for Stand and Deliver, Good Will Hunting, and Contact. 8 students vote for A Beautiful Mind and Proof. 8 students vote for Stand and Deliver and Proof. 8 students vote for Good Will Hunting and Contact. 1 student votes for Contact. Which film will be chosen for the last day of class? Alabama Or Not Alabama, That Is The Question. 30. Twenty-seven digital projectors are to be apportioned to four schools based on their enrollments. Campus A B C D Total Enrollment 840 1936 310 2744 5830 Hamilton 4 9 1 13 27 a) If the number of projectors increases from 27 to 28, find the new Hamilton apportionment. Will the Alabama paradox occur? b) If the number of projectors increases from 28 to 29, find the new Hamilton apportionment. Will the Alabama paradox occur?

A Closely Guarded Paradox? 31. A college apportions 40 security guards among three campuses according to their enrollments. Campus A B C Total Enrollment 356 1054 2590 4000 a) Find the Hamilton apportionment for the security guards. b) After one semester, the campuses have the following new enrollments. Campus A B C Total Enrollment 370 1079 2600 4049 Find the new Hamilton apportionment. Has the population paradox occurred? New States Or New Cities Paradox? 32. A company has offices in Boston and Chicago. Twenty-two vice-presidents are to be apportioned to the offices based upon the number of employees. Office Boston Chicago Total Employees 151 1210 1361 a) Find the Hamilton apportionment of the vice-presidents. b) The company opens a new office in San Francisco, and decides on a new total of 24 vicepresidents. Office Boston Chicago San Francisco Total Employees 151 1210 135 1496 Find the new Hamilton apportionment. Has the new states paradox occurred?