Chapter 13

Describe the majority criterion.

Suppose that the pairwise comparison method is used to determine the winner in an election. If there are six candidates, how many comparisons must be made?

Describe the head-to-head criterion.

Suppose that the pairwise comparison method is used to determine the winner in an election. If there are eight candidates, how many comparisons must be made?

A hospital has a nursing staff of 250 nurses working in four shifts: \(A\) (7:00 A.M. to 1:00 P.M.), B (1:00 P.M. to 7:00 P.M.), \(C\) (7:00 P.M. to 1:00 A.M.), and D (1:00 A.M. to 7:00 A.M.). The number of nurses apportioned to each shift is based on the average number of patients per shift, given in the following table. Use this information to solve. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Shift } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline \begin{array}{l} \text { Average Number } \\ \text { of Patients } \end{array} & 453 & 650 & 547 & 350 \\ \hline \end{array} $$ Use Webster's method to apportion the 250 nurses among the shifts in the hospital.

Describe the monotonicity criterion.

Suppose that the pairwise comparison method is used to determine the winner in an election. If there are nine candidates, how many comparisons must be made?

The table shows the 1790 United States census. In 1793, at th direction of President George Washington, 105 seats in t House of Representatives were to be divided among the 15 stat according to their 1790 populations. Use this information to sol Exercises 23-26. $$ \begin{aligned} &1790 \text { UNITED STATES CENSUS }\\\ &\begin{array}{|l|r|l|r|} \hline \text { Connecticut } & 236,841 & \text { New York } & 331,589 \\ \hline \text { Delaware } & 55,540 & \text { North Carolina } & 353,523 \\ \hline \text { Georgia } & 70,835 & \text { Pennsylvania } & 432,879 \\ \hline \text { Kentucky } & 68,705 & \text { Rhode Island } & 68,446 \\ \hline \text { Maryland } & 278,514 & \text { South Carolina } & 206,236 \\ \hline \text { Massachusetts } & 475,327 & \text { Vermont } & 85,533 \\ \hline \text { New Hampshire } & 141,822 & \text { Virginia } & 630,560 \\ \hline \text { New Jersey } & 179,570 & & \\ \hline \end{array} \end{aligned} $$ Use Hamilton's method to find each state's apportionment of congressional seats.

Describe how to find a standard divisor.

In Exercises 28–31, determine whether each statement makes sense or does not make sense, and explain your reasoning. My candidate received a majority of first-place votes and lost the election.

Voters are asked to rank four brands of soup: \(A, B, C\), and D. The votes are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{3 4} & \mathbf{3 0} & \mathbf{6} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { D } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { D } & \text { B } & \text { C } \\ \hline \text { Fourth Choice } & \text { D } & \text { A } & \text { A } & \text { A } \\ \hline \end{array} $$ Determine the winner using the Borda count method.

How is the lower quota found from a standard quota?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My candidate was favored when compared head-to-head with every other candidate and lost the election.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A candidate won an election and, in a reelection, the only changes were changes that favored the candidate, so I'm certain that this candidate won the reelection.

Describe the apportionment problem.

In your own words, describe Hamilton's method of apportionment.

What is the quota rule?

The programmers at the Theater Channel need to select a live musical to introduce their new network. The five choices are Cabaret \((C)\), The Producers \((P)\), Rent \((R)\), Sweeney Todd (S), or West Side Story \((W)\). The 22 programmers rank their choices, summarized in the following preference table. Use the table to solve.$ $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{5} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { C } & \text { S } & \text { C } & \text { W } & \text { W } & \text { P } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { P } & \text { P } & \text { R } & \text { S } \\ \hline \text { Third Choice } & \text { P } & \text { W } & \text { R } & \text { R } & \text { S } & \text { C } \\ \hline \text { Fourth Choice } & \text { W } & \text { P } & \text { S } & \text { S } & \text { C } & \text { R } \\ \hline \text { Fifth Choice } & \text { S } & \text { C } & \text { W } & \text { C } & \text { P } & \text { W } \\ \hline \end{array} $$ Determine which musical is selected using the pairwise comparison method.

Explain why Hamilton's method satisfies the quota rule.

Construct a preference table for an election among three candidates, \(A, B\), and \(C\), with the given characteristics. Do not use any of the tables from this section. The winner by the plurality method violates the irrelevant alternatives criterion.

Describe the difference between how modified quotas are rounded using Jefferson's method and Adams's method.

Construct a preference table for an election among three candidates, \(A, B\), and \(C\), with the given characteristics. Do not use any of the tables from this section. The winner by the plurality-with-elimination method violates the monotonicity criterion.

Citizen-initiated ballot measures often present voters with controversial issues over which they do not think alike. Here's one your author would like to initiate: Please rank each of the following options regarding permitting dogs on national park trails. i. Unleashed dogs accompanied by their caregivers should be permitted on designated national park trails. ii. Leashed dogs accompanied by their caregivers should be permitted on designated national park trails. iii. No dogs should be permitted on any national park trails. Your author was not happy with the fact that he could not take his dog running with him on the park trails at Point Reyes National Seashore. Of course, that is his issue. For this project, group members should write a ballot measure, perhaps controversial, like the sample above, but dealing with an issue of relevance to your campus and community. Rather than holding an election, use a random sample of students on your campus, administer the ballot, and have them rank their choices. a. Use each of the four voting methods to determine the winning option for your ballot measure. b. Check to see if any of the four fairness criteria are violated.

Describe the difference between the modified divisor, \(d\), in terms of the standard divisor using Jefferson's method and Adams's method.

Three candidates, A, B, and C, are running for mayor. Election rules stipulate that the plurality method will determine the winner. In the event that the plurality method leads to a tie, the Borda count method will decide the winner. The election results are summarized in the following preference table. Under these rules, which candidate becomes the new mayor? $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 , 0 0 0} & \mathbf{7 5 0 0} & \mathbf{4 5 0 0} \\ \hline \text { First Choice } & \text { C } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$

In allocating congressional seats, how does Hamilton's method choose some states over others for preferential treatment? Explain how this is avoided in Jefferson's and Adams's methods.

Three candidates, A, B, and \(\mathrm{C}\), are running for mayor. Election rules stipulate that the pairwise comparison method will determine the winner. In the event that the pairwise comparison method leads to a tie, the Borda count method will decide the winner. The election results are summarized in the following preference table. Under these rules, which candidate becomes the new mayor? $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{6 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} \\ \hline \text { First Choice } & \text { A } & \text { C } & \text { B } & \text { A } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { B } & \text { A } & \text { B } & \text { A } \\ \hline \end{array} $$

How are modified quotas rounded using Webster's method?

What is a preference ballot?

Describe what is contained in a preference table. What does the table show?

In this Exercise Set, we have used apportionment methods to divide congressional seats, assign computers to schools, assign doctors to clinics, divide police officers among precincts, divide shares of stock, assign sections of bilingual math, assign buses to city routes, and assign nurses to hospital shifts. Describe another situation that requires the use of apportionment methods.

Describe the plurality method. Why is ranking not necessary when using this method?

Describe the Borda count method. Is it possible to use this method without ranking the candidates? Explain.

What is the plurality-with-elimination method? Why is it advantageous to rank the candidates when using this method?

What is the pairwise comparison method? Is it possible to use this method without ranking the candidates? Explain.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A modified quota of \(46.01\) and a final apportionment of 47 is a violation of the quota rule.

Describe the process used to determine how many comparisons must be made with the pairwise comparison method.

Why is it important to choose a voting system before an election takes place?

Playwright Tom Stoppard wrote, "It's not the voting that's democracy; it's the counting." Explain what he meant by this.

A small country is composed of three states, \(A, B\), and \(C\). The country's constitution specifies that congressional seats will be divided among the three states according to their respective populations. Write an apportionment problem satisfying the given criterion. Hamilton's method and Adams's method result in the same apportionment.

In Exercises 49-52, determine whether each statement makes sense or does not make sense, and explain your reasoning. A candidate has a majority of the vote, yet lost the election using the plurality method.

The method currently used to apportion the U.S. House of Representatives is known as the Huntington-Hill method, and more commonly as the method of equal proportions. Research and present a group report on this method. Include the history of how the method came into use and describe how the method works.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A candidate has a majority of the vote, yet lost the election using the plurality-with-elimination method.

Research and present a group report on a brief history of apportionment in the United States.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A candidate has a plurality of the vote, yet lost the election using the Borda count method.

A candidate has a plurality of the vote, yet lost the election using the Borda count method.A candidate won the election using the plurality-withelimination method, yet lost the election when the votes were counted by the pairwise comparison method.

Construct a preference table for an election with candidates \(A, B\), and \(C\) satisfying the given condition. C wins using the pairwise comparison method.

Research and present a group report on how voting is conducted for the Academy Awards. Describe the single transferable voting method, a variation of the pluralitywith-elimination method, in the nomination stage for best picture. (Members of the Irish Senate are also elected by this method.) Be sure to describe some of the more bizarre occurrences at the Oscar ceremonies.

Research and present a group report on how voting is conducted for one or more of the following awards: the Heisman Trophy, the Nobel Prize, the Grammy, the Tony, the Emmy, the Pulitzer Prize, or any event or award that the group finds intriguing. Be sure to discuss how the nominees are selected, who participates in the voting, the voting system or systems used, and who counts the results.