Chapter 13

A small country is comprised of four states, \(A, B, C\), and \(D\). The population of each state, in thousands, is given in the following table. Use this information to solve Exercises 1-4. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 138 & 266 & 534 & 662 & 1600 \\ \hline \end{array} $$ According to the country's constitution, the congress wil have 80 seats, divided among the four states according their respective populations. a. Find the standard divisor, in thousands. How many people are there for each seat in congress? b. Find each state's standard quota. c. Find each state's lower quota and upper quota.

Voters in a small town are considering four proposals, A, B, \(C\), and D, for the design of affordable housing. The winning design is to be determined by the Borda count method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{3 0 0} & \mathbf{1 2 0} & \mathbf{9 0} & \mathbf{6 0} \\ \hline \text { First Choice } & \text { D } & \text { C } & \text { C } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { A } & \text { A } & \text { D } \\ \hline \text { Third Choice } & \text { B } & \text { B } & \text { D } & \text { B } \\ \hline \text { Fourth Choice } & \text { C } & \text { D } & \text { B } & \text { C } \\ \hline \end{array} $$ a. Which design has a majority of first-place votes? b. Using the Borda count method, which design will be used for the affordable housing? c. Is the majority criterion satisfied? Explain your answer.

In Exercises 1-2, the preference ballots for three candidates \((A, B\), and \(C)\) are shown. Fill in the number of votes in the first row of the given preference table. ABC BCA BCA CBA CBA ABC ABC BCA BCA CBA ABC ABC $$ \begin{array}{|l|l|l|l|} \hline \text { Number of Votes } & & & \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ BCA ABC ABC CBA

A school district has 57 new laptop computers to be divided among four schools, according to their respective enrollments. The table shows the number of students enrolled in each school. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \text { Enrollment } & 5040 & 4560 & 4040 & 610 & 14,250 \\ \hline \end{array} $$ a. Apportion the laptop computers using Hamilton's method. b. Use Hamilton's method to determine if the Alabama paradox occurs if the number of laptop computers is increased from 57 to 58 . Explain your answer.

A small country is comprised of four states, \(A, B, C\), and \(D\). The population of each state, in thousands, is given in the following table. Use this information to solve. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 138 & 266 & 534 & 662 & 1600 \\ \hline \end{array} $$ According to the country's constitution, the congress will have 200 seats, divided among the four states according to their respective populations. a. Find the standard divisor, in thousands. How many people are there for each seat in congress? b. Find each state's standard quota. c. Find each state's lower quota and upper quota.

Fifty-three people are asked to taste-test and rank three different brands of yogurt, \(A, B\), and \(C\). The preference table shows the rankings of the 53 voters. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{2 7} & \mathbf{2 4} & \mathbf{2} \\\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Which brand has a majority of first-place votes? b. Suppose that the Borda count method is used to determine the winner. Which brand wins the taste test? c. Is the majority criterion satisfied? Explain your answer.

The table shows the populations of three states in a country with a population of 20,000 . Use Hamilton's method to show that the Alabama paradox occurs if the number of seats in congress is increased from 40 to 41 . $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \text { Population } & 680 & 9150 & 10,170 & 20,000 \\ \hline \end{array} $$

MTV's Real World is considering three cities for its new season: Amsterdam (A), Rio de Janeiro (R), or Vancouver (V). Programming executives and the show's production team vote to decide where the new season will be taped. The winning city is to be determined by the plurality method. The preference table for the election is shown at the top of the next column. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2} & \mathbf{9} & \mathbf{4} & \mathbf{4} \\ \hline \text { First Choice } & \text { A } & \text { V } & \text { V } & \text { R } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { A } & \text { A } \\ \hline \text { Third Choice } & \text { V } & \text { A } & \text { R } & \text { V } \\ \hline \end{array} $$ a. Which city is favored over all others using a head-tohead comparison? b. Which city wins the vote using the plurality method? c. Is the head-to-head criterion satisfied? Explain your answer.

The table at the top of the next column shows the populations, in thousands, of the three states in a country with a population of 3760 thousand. Use Hamilton's method to show that the Alabama paradox occurs if the number of seats in congress is increased from 24 to 25 . $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 530 & 990 & 2240 & 3760 \\ \hline \end{array} $$

A computer company is considering opening a new branch in Atlanta (A), Boston (B), or Chicago (C). Senior managers vote to decide where the new branch will be located. The winning city is to be determined by the plurality method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{2 0} & \mathbf{1 9} & \mathbf{5} \\\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Which city is favored over all others using a head-tohead comparison? b. Which city wins the vote using the plurality method? c. Is the head-to-head criterion satisfied? Explain your answer.

A small country has 24 seats in the congress, divided among the three states according to their respective populations. The table shows each state's population, in thousands, before and after the country's population increase. $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \begin{array}{l} \text { Original Population } \\ \text { (in thousands) } \end{array} & 530 & 990 & 2240 & 3760 \\ \hline \begin{array}{l} \text { New Population } \\ \text { (in thousands) } \end{array} & 680 & 1250 & 2570 & 4500 \\ \hline \end{array} $$ a. Use Hamilton's method to apportion the 24 congressional seats using the original population. b. Find the percent increase, to the nearest tenth of a percent, in the population of each state. c. Use Hamilton's method to apportion the 24 congressional seats using the new population. Does the population paradox occur? Explain your answer.

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Human- } \\ \text { ities } \end{array} & \begin{array}{c} \text { Social } \\ \text { Science } \end{array} & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Business } & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} \\ \hline \text { Enrollment } & 1050 & 1410 & 1830 & 2540 & 3580 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

A town is voting on an ordinance dealing with smoking in public spaces. The options are (A) permit unrestricted smoking; (B) permit smoking in designated areas only; and \((C)\) ban all smoking in public places. The winner is to be determined by the Borda count method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 0} & \mathbf{6 0} & \mathbf{3 0} & \mathbf{3 0} & \mathbf{3 0} \\ \hline \text { First Choice } & \text { A } & \text { C } & \text { B } & \text { C } & \text { B } \\ \hline \text { Second Choice } & \text { C } & \text { B } & \text { A } & \text { A } & \text { C } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { C } & \text { B } & \text { A } \\ \hline \end{array} $$ a. Which option is favored over all others using a head-tohead comparison? b. Which option wins the vote using the Borda count method? c. Is the head-to-head criterion satisfied? Explain your answer.

Your class is given the option of choosing a day for the final exam. The students in the class are asked to rank the three available days, Monday (M), Wednesday (W), and Friday (F). The results of the election are shown in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{8} & \mathbf{3} & \mathbf{1} \\ \hline \text { First Choice } & \mathrm{F} & \mathrm{F} & \mathrm{W} & \mathbf{M} \\ \hline \text { Second Choice } & \mathrm{W} & \mathrm{M} & \mathrm{F} & \mathrm{W} \\ \hline \text { Third Choice } & \mathrm{M} & \mathrm{W} & \mathrm{M} & \mathrm{F} \\ \hline \end{array} $$ a. How many students voted in the election? b. How many students selected the days in this order: \(\mathrm{F}, \mathrm{M}, \mathrm{W} ?\) c. How many students selected Friday as their first choice for the final? d. How many students selected Wednesday as their first choice for the final?

A country has 200 seats in the congress, divided among the five states according to their respective populations. The table shows each state's population, in thousands, before and after the country's population increase. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { Total } \\ \hline \begin{array}{l} \text { Original } \\ \text { Population } \\ \text { (in thousands) } \end{array} & 2224 & 2236 & 2640 & 3030 & 9870 & 20,000 \\ \hline \begin{array}{l} \text { New Population } \\ \text { (in thousands) } \end{array} & 2424 & 2436 & 2740 & 3130 & 10,070 & 20,800 \\ \hline \end{array} $$ a. Use Hamilton's method to apportion the 200 congressional seats using the original population. b. Find the percent increase, to the nearest tenth of a percent, in the population of each state. c. Use Hamilton's method to apportion the 200 congressional seats using the new population. What do you observe about the percent increases for states A and \(B\) and their respective changes in apportioned seats? Is this the population paradox?

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Liberal } \\ \text { Arts } \end{array} & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} & \text { Business } & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Sciences } \\ \hline \text { Enrollment } & 1180 & 1290 & 2140 & 2930 & 3320 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

A town is voting on an ordinance dealing with nudity at its public beaches. The options are (A) make clothing optional at all beaches; (B) permit nudity at designated beaches only; and (C) permit no nudity at public beaches. The winner is to be determined by the Borda count method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{2 0 0} & \mathbf{8 0} & \mathbf{8 0} \\ \hline \text { First Choice } & \text { C } & \text { B } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { A } & \text { B } \\ \hline \text { Third Choice } & \text { B } & \text { C } & \text { C } \\ \hline \end{array} $$ a. Which option is favored over all others using a head-tohead comparison? b. Which option wins the vote using the Borda count method? c. Is the head-to-head criterion satisfied? Explain your answer.

Students at your college are given the option of choosing a topic for which a speaker will be selected. Students are asked to rank three topics: Technology (T), Environmental Issues (E), and Terrorism in the Name of Religion (R). The results of the election are shown in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{7 0} & \mathbf{3 0} & \mathbf{1 0} & \mathbf{5} \\ \hline \text { First Choice } & \mathrm{R} & \mathrm{T} & \mathrm{T} & \mathrm{E} \\ \hline \text { Second Choice } & \mathrm{E} & \mathrm{R} & \mathrm{E} & \mathrm{T} \\ \hline \text { Third Choice } & \mathrm{T} & \mathrm{E} & \mathrm{R} & \mathrm{R} \\ \hline \end{array} $$ a. How many students voted? b. How many students selected the topics in this order: \(\mathrm{T}, \mathrm{E}, \mathrm{R} ?\) c. How many students selected technology as their first choice for a speaker's topic? d. How many students selected environmental issues as their second choice for a speaker's topic?

A town has 40 mail trucks and four districts in which mail is distributed. The trucks are to be apportioned according to each district's population. The table shows these populations before and after the town's population increase. Use Hamilton's method to show that the population paradox occurs. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { District } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \text { Original Population } & 1188 & 1424 & 2538 & 3730 & 8880 \\ \hline \text { New Population } & 1188 & 1420 & 2544 & 3848 & 9000 \\ \hline \end{array} $$

A small country is composed of five states, \(A, B, C, D\), and \(E\). The population of each state is given in the following table. Congress will have 57 seats, divided among the five states according to their respective populations. Use Jefferson's method with \(d=32,920\) to apportion the 57 congressional seats. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Population } & 126,316 & 196,492 & 425,264 & 526,664 & 725,264 \\\ \hline \end{array} $$

The following preference table gives the results of a straw vote among three candidates, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 0} & \mathbf{8} & \mathbf{7} & \mathbf{4} \\ \hline \text { First Choice } & \text { C } & \text { B } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { C } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, which candidate wins the straw vote? b. In the actual election, the four voters in the last column who voted \(A, C, B\), in that order, change their votes to \(\mathrm{C}, \mathrm{A}, \mathrm{B}\). Using the plurality-with-elimination method, which candidate wins the actual election? c. Is the monotonicity criterion satisfied? Explain your answer.

A town has five districts in which mail is distributed and 50 mail trucks. The trucks are to be apportioned according to each district's population. The table shows these populations before and after the town's population increase. Use Hamilton's method to show that the population paradox occurs. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { District } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { Total } \\ \hline \begin{array}{l} \text { Original } \\ \text { Population } \end{array} & 780 & 1500 & 1730 & 2040 & 2950 & 9000 \\ \hline \text { New Population } & 780 & 1500 & 1810 & 2040 & 2960 & 9090 \\ \hline \end{array} $$

A small country is comprised of four states, A, B, C, and D. The population of each state, in thousands, is given in the following table. Congress will have 400 seats, divided among the four states according to their respective populations. Use Jefferson's method with \(d=7.82\) to apportion the 400 congressional seats. $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 424 & 664 & 892 & 1162 \\ \hline \end{array} $$

The preference table gives the results of a straw vote among three candidates, A, B, and C. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{1 2} & \mathbf{1 0} & \mathbf{6} \\ \hline \text { First Choice } & \text { C } & \text { B } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { C } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, which candidate wins the straw vote? b. In the actual election, the six voters in the last column who voted \(\mathrm{A}, \mathrm{C}, \mathrm{B}\), in that order, change their votes to C, A, B. Using the plurality-with-elimination method, which candidate wins the actual election? c. Is the monotonicity criterion satisfied? Explain your answer.

The travel club members are voting for the American city they will visit next semester: New York (N), San Francisco (S), or Chicago (C). Their votes are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 6} & \mathbf{8} & \mathbf{6} & \mathbf{4} \\ \hline \text { First Choice } & \text { S } & \text { N } & \text { N } & \text { C } \\ \hline \text { Second Choice } & \text { N } & \text { S } & \text { C } & \text { N } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { S } & \text { S } \\ \hline \end{array} $$ Which city is selected using the plurality method?

A corporation has two branches, A and B. Each year the company awards 100 promotions within its branches. The table shows the number of employees in each branch. $$ \begin{array}{|l|c|c|c|} \hline \text { Branch } & \text { A } & \text { B } & \text { Total } \\ \hline \text { Employees } & 1045 & 8955 & 10,000 \\ \hline \end{array} $$ a. Use Hamilton's method to apportion the promotions. b. Suppose that a third branch, C, with the number of employees shown in the table below, is added to the corporation. The company adds five new yearly promotions for branch C. Use Hamilton's method to determine if the new-states paradox occurs when the promotions are reapportioned.

An HMO has 150 doctors to be apportioned among four clinics. The HMO decides to apportion the doctors based on the average weekly patient load for each clinic, given in the following table. Use Jefferson's method to apportion the 150 doctors. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Clinic } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \begin{array}{l} \text { Average Weekly } \\ \text { Patient Load } \end{array} & 1714 & 5460 & 2440 & 5386 \\ \hline \end{array} $$

Members of the Student Activity Committee at a college are considering three film directors to speak at a campus arts festival: Ron Howard (H), Spike Lee (L), and Steven Spielberg (S). Committee members vote for their preferred speaker. The winner is to be selected by the pairwise comparison method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 0} & \mathbf{8} & \mathbf{5} \\ \hline \text { First Choice } & \text { H } & \text { L } & \text { S } \\ \hline \text { Second Choice } & \text { S } & \text { S } & \text { L } \\ \hline \text { Third Choice } & \text { L } & \text { H } & \text { H } \\ \hline \end{array} $$ a. Using the pairwise comparison method, who is selected as the speaker? b. Prior to the announcement of the speaker, Ron Howard informs the committee that he will not be able to participate due to other commitments. Construct a new preference table for the election with \(\mathrm{H}\) eliminated. Using the new table and the pairwise comparison method, who is selected as the speaker? c. Is the irrelevant alternatives criterion satisfied? Explain your answer.

A corporation has three branches, A, B, and C. Each year the company awards 60 promotions within its branches. The table shows the number of employees in each branch. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Branch } & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \text { Employees } & 209 & 769 & 2022 & 3000 \\ \hline \end{array} $$ a. Use Hamilton's method to apportion the promotions. b. Suppose that a fourth branch, D, with the number of employees shown in the table below, is added to the corporation. The company adds five new yearly promotions for branch D. Use Hamilton's method to determine if the new-states paradox occurs when the promotions are reapportioned. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Branch } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \text { Employees } & 209 & 769 & 2022 & 260 & 3260 \\ \hline \end{array} $$

An HMO has 70 doctors to be apportioned among six clinics. The HMO decides to apportion the doctors based on the average weekly patient load for each clinic, given in the following table. Use Jefferson's method to apportion the 70 doctors. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Clinic } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \hline \begin{array}{l} \text { Average Weekly } \\ \text { Patient Load } \end{array} & 316 & 598 & 396 & 692 & 426 & 486 \\ \hline \end{array} $$

Members of the Student Activity Committee at a college are considering three actors to speak at a campus festival on women in the arts: Whoopi Goldberg \((\mathrm{G})\), Julia Roberts \((\mathrm{R})\), and Meryl Streep (S). Committee members vote for their preferred speaker. The winner is to be selected by the pairwise comparison method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2} & \mathbf{8} & \mathbf{6} \\ \hline \text { First Choice } & \text { S } & \text { R } & \text { G } \\ \hline \text { Second Choice } & \text { G } & \text { G } & \text { R } \\ \hline \text { Third Choice } & \text { R } & \text { S } & \text { S } \\ \hline \end{array} $$ a. Using the pairwise comparison method, who is selected as the speaker? b. Prior to the announcement of the speaker, Meryl Streep informs the committee that she will not be able to participate due to other commitments. Construct a new preference table for the election with \(S\) eliminated. Using the new table and the pairwise comparison method, who is selected as the speaker? c. Is the irrelevant alternatives criterion satisfied? Explain your answer.

a. A country has two states, state A, with a population of 9450 , and state B, with a population of 90,550 . The congress has 100 seats, divided between the two states according to their respective populations. Use Hamilton's method to apportion the congressional seats to the states. b. Suppose that a third state, state \(\mathrm{C}\), with a population of 10,400 , is added to the country. The country adds 10 new congressional seats for state C. Use Hamilton's method to show that the new-states paradox occurs when the congressional seats are reapportioned.

In Exercises 11-18, the preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{2 0} & \mathbf{1 6} & \mathbf{1 0} & \mathbf{4} \\ \hline \text { First Choice } & \text { D } & \text { C } & \text { C } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { A } & \text { B } & \text { B } \\ \hline \text { Third Choice } & \text { B } & \text { B } & \text { D } & \text { D } \\ \hline \text { Fourth Choice } & \text { C } & \text { D } & \text { A } & \text { C } \\ \hline \end{array} $$ a. Using the Borda count method, who is the winner? b. Is the majority criterion satisfied? Explain your answer.

a. A country has three states, state A, with a population of 99,000 , state B, with a population of 214,000 , and state C, with a population of 487,000 . The congress has 50 seats, divided among the three states according to their respective populations. Use Hamilton's method to apportion the congressional seats to the states. b. Suppose that a fourth state, state D, with a population of 116,000 , is added to the country. The country adds seven new congressional seats for state D. Use Hamilton's method to show that the new-states paradox occurs when the congressional seats are reapportioned.

Four people pool their money to buy 60 shares of stock. The amount that each person contributes is shown in the following table. Use Adams's method with \(d=108\) to apportion the shares of stock. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Person } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Contribution } & \$ 2013 & \$ 187 & \$ 290 & \$ 3862 \\ \hline \end{array} $$

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{2 0} & \mathbf{1 5} & \mathbf{3} & \mathbf{1} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { D } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { D } & \text { B } & \text { C } \\ \hline \text { Fourth Choice } & \text { D } & \text { A } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Using the Borda count method, who is the winner? b. Is the majority criterion satisfied? Explain your answer.

What is the Alabama paradox?

What is the population paradox?

Twenty sections of bilingual math courses, taught in both English and Spanish, are to be offered in introductory algebra, intermediate algebra, and liberal arts math. The preregistration figures for the number of students planning to enroll in these bilingual sections are given in the following table. Use Webster's method with \(d=29.6\) to determine how many bilingual sections of each course should be offered. $$ \begin{array}{|l|c|c|c|} \hline \text { Course } & \begin{array}{c} \text { Introductory } \\ \text { Algebra } \end{array} & \begin{array}{c} \text { Intermediate } \\ \text { Algebra } \end{array} & \begin{array}{c} \text { Liberal Arts } \\ \text { Math } \end{array} \\ \hline \text { Enrollment } & 130 & 282 & 188 \\ \hline \end{array} $$

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{8} & \mathbf{4} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { D } & \text { A } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { C } \\ \hline \text { Fourth Choice } & \text { D } & \text { A } & \text { B } \\ \hline \end{array} $$ a. Using the Borda count method, who is the winner? b. Is the majority criterion satisfied? Explain your answer. c. Is the head-to-head criterion satisfied? Explain your answer. d. Suppose that candidate \(C\) drops out of the race. Using the Borda count method, who among the remaining candidates wins the election? Is the irrelevant alternatives criterion satisfied? Explain your answer.

What is the new-states paradox?

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{1 2} & \mathbf{1 0} & \mathbf{6} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { A } & \text { B } \\ \hline \text { Fourth Choice } & \text { D } & \text { D } & \text { D } & \text { A } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, who is the winner? b. The six voters on the right all move candidate A from last place on their preference lists to first place on their preference lists. Construct a new preference table for the election. Using this table and the plurality-with- elimination method, who is the winner? Is the monotonicity criterion satisfied? Explain your answer.

According to Balinski and Young's Impossibility Theorem, can the democratic ideal of "one person, one vote" ever be perfectly achieved? Explain your answer.

A rapid transit service operates 200 buses along five routes, \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\), and \(\mathrm{E}\). The number of buses assigned to each route is based on the average number of daily passengers per route, given in the following table. Use Webster's method to apportion the buses. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Route } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \begin{array}{l} \text { Average Number } \\ \text { of Passengers } \end{array} & 1087 & 1323 & 1592 & 1596 & 5462 \\ \hline \end{array} $$

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 6} & \mathbf{1 4} & \mathbf{1 2} & \mathbf{4} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { D } & \text { D } & \text { C } & \text { E } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { B } & \text { A } & \text { A } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { E } & \text { B } & \text { D } \\ \hline \text { Fourth Choice } & \text { D } & \text { C } & \text { C } & \text { D } & \text { B } \\ \hline \text { Fifth Choice } & \text { E } & \text { E } & \text { A } & \text { E } & \text { C } \\ \hline \end{array} $$ a. Using the Borda count method, who is the winner? b. Is the majority criterion satisfied? Explain your answer. c. Is the head-to-head criterion satisfied? Explain your answer.

In Exercises 18-21, determine whether each statement makes sense or does not make sense, and explain your reasoning. The county hired seven new doctors to apportion among its three clinics. Although our local clinic has the same proportion of the county's patients as it did before the doctors were hired, it now has one fewer doctor.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The population of state A grew at a faster rate than that of state B, yet state A lost an apportioned seat in the legislature to state \(B\).

The preference table shows the results of an election among three candidates, A, B, and C. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Using the plurality method, who is the winner? b. Is the majority criterion satisfied? Explain your answer. c. Is the head-to-head criterion satisfied? Explain your answer. d. The two voters on the right both move candidate A from last place on their preference lists to first place on their preference lists. Construct a new preference table for the election. Using the table and the plurality method, who is the winner? e. Suppose that candidate \(\mathrm{C}\) drops out, but the winner is still chosen by the plurality method. Is the irrelevant alternatives criterion satisfied? Explain your answer. f. Do your results from parts (b) through (e) contradict Arrow's Impossibility Theorem? Explain your answer.

In Exercises 19-22, suppose that the pairwise comparison method is used to determine the winner in an election. If there are five candidates, how many comparisons must be made?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Balinski and Young's Theorem shows that "one person, one vote" is mathematically possible.