$\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm {~b} > \mathrm { a } > \mathrm { c } > \mathrm { d } \\ 11 & \mathrm { c } > \mathrm { a } > \mathrm { d } > \mathrm { b } \\ 9 & \mathrm {~d} > \mathrm { c } > \mathrm { a } > \mathrm { b } \\ 5 & \mathrm { a } > \mathrm { d } > \mathrm { c } > \mathrm { b } \\ 1 & \mathrm { c } > \mathrm { d } > \mathrm { a } > \mathrm { b } \end{array}$ -Construct a voter profile for 41 voters and 4 candidates that has a Condorcet Candidate that fails to be elected by both the Borda Method and the Hare Method.

To construct a voter profile for 41 vote

Suppose 8 candidates a, b, c, d, e, f, g, h are involved in a Pairwise Comparison Method election. The number of pairwise comparisons each candidate wins is shown in the table. \[\begin{array} { l | l l l l l l l l l } \text { Candidate } & \mathrm { a } & \mathrm { b } & \mathrm { c } & \mathrm { d } & \mathrm { e } & \mathrm { f } & \mathrm { g } & \mathrm { h } \\ \hline \text { Number of Pairwise Comparisons Won } & 4 & 5 & ? & 3 & 4 & 2 & 1 & 4 \end{array}\] How many comparisons does candidate c win?

A) 5 B) 4 C) 3 D) 6 A) 5 B) 4 C) 3 D) 6

For the voter profile given below, who (or what)is selected if a runoff election is held between the candidates that rank second and third; and the winner of that election faces the candidate with the Most first place votes, to decide the final outcome of the election? $\begin{array}{c|l} \text { Number of Voters } & \text { Ranking } \\ \hline 2 & \mathrm{c}>\mathrm{j}>\mathrm{t} \\ 3 & \mathrm{c}>\mathrm{t}>\mathrm{j} \\ 3 & \mathrm{j}>\mathrm{c}>\mathrm{t} \\ 1 & \mathrm{j}>\mathrm{t}>\mathrm{c} \\ 5 & \mathrm{t}>\mathrm{c}>\mathrm{j} \\ 1 & \mathrm{t}>\mathrm{j}>\mathrm{c} \end{array}$

A) $j$ B) $t$ C) $\mathrm { c }$ A) $j$ B) $t$ C) $\mathrm { c }$

Solve the problem. -The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with Candidates Clark (C), Jones (J), and Smith (S). $\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 7 & \mathrm {~J} > \mathrm { S } > \mathrm { C } \\ 10 & \mathrm {~S} > \mathrm { J } > \mathrm { C } \\ 4 & \mathrm {~J} > \mathrm { C } > \mathrm { S } \\ 8 & \mathrm { C } > \mathrm { S } > \mathrm { J } \end{array}$ Determine the winner using the pairwise comparison method and state whether the winner Received a majority of votes.

A) Jones; no B) Jones; yes C) Smith; yes D) Smith; no A) Jones; no B) Jones; yes C) Smith; yes D) Smith; no

Central City Community College is made up of four divisions: Liberal Arts, Sciences, Business, and Humanities. The faculty senate has 50 senators to be apportioned based on the faculty of each Division as shown below. $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Faculty } & 130 & 175 & 46 & 49 \end{array}$ Use the Adams Method to apportion the senators.

A) $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 17 & 21 & 6 & 6 \end{array}$ B) $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 16 & 22 & 6 & 6 \end{array}$ C) $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 16 & 21 & 6 & 7 \end{array}$ D) $\begin{array}{l|ccccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 15 & 23 & 6 & 6 \end{array}$ A) $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 17 & 21 & 6 & 6 \end{array}$ B) $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 16 & 22 & 6 & 6 \end{array}$ C) $\begin{array}{l|cccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 16 & 21 & 6 & 7 \end{array}$ D) $\begin{array}{l|ccccc} \text { Division } & \text { Liberal Arts } & \text { Sciences } & \text { Business } & \text { Humanities } \\ \hline \text { Number of Senators } & 15 & 23 & 6 & 6 \end{array}$

Solve the problem. -The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with Candidates Clark (C), Jones (J), and Smith (S). $\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 7 & \mathrm {~J} > \mathrm { S } > \mathrm { C } \\ 10 & \mathrm {~S} > \mathrm { J } > \mathrm { C } \\ 4 & \mathrm {~J} > \mathrm { C } > \mathrm { S } \\ 8 & \mathrm { C } > \mathrm { S } > \mathrm { J } \end{array}$ Determine the winner using the plurality method and state whether the winner received a majority Of votes.

A) Clark; yes B) Jones; no C) Smith; yes D) Clark; no A) Clark; yes B) Jones; no C) Smith; yes D) Clark; no

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to The following voter profile. $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 5 & \mathrm{~A}>\mathrm{H}>\mathrm{F}>\mathrm{S} \\ 7 & \mathrm{H}>\mathrm{S}>\mathrm{A}>\mathrm{F} \\ 4 & \mathrm{~F}>\mathrm{S}>\mathrm{H}>\mathrm{A} \\ 7 & \mathrm{~A}>\mathrm{F}>\mathrm{S}>\mathrm{H} \\ 5 & \mathrm{~A}>\mathrm{H}>\mathrm{S}>\mathrm{F} \\ 6 & \mathrm{H}>\mathrm{F}>\mathrm{S}>\mathrm{A} \\ 8 & \mathrm{~F}>\mathrm{S}>\mathrm{A}>\mathrm{H} \end{array}$ Determine the winner using the pairwise comparison method.

A) Florida & Hawaii (tie) B) Alaska C) Alaska & Hawaii (tie) D) Alaska & Florida (tie) A) Florida & Hawaii (tie) B) Alaska C) Alaska & Hawaii (tie) D) Alaska & Florida (tie)

Write a voter profile for the voting situation. -Ten voters are asked to rank 4 brands of cell phones: A, B, C, and D. The ten voters turn in the following ballots showing their preferences in order: $\begin{array}{l|llllllllll} \text { Voter } & \# 1 & \# 2 & \# 3 & \# 4 & \# 5 & \# 6 & \# 7 & \# 8 & \# 9 & \# 10 \\ \hline \text { 1st place } & \mathrm{A} & \mathrm{B} & \mathrm{B} & \mathrm{B} & \mathrm{A} & \mathrm{B} & \mathrm{B} & \mathrm{B} & \mathrm{B} & \mathrm{A} \\ \hline \text { 2nd place } & \mathrm{B} & \mathrm{C} & \mathrm{A} & \mathrm{C} & \mathrm{B} & \mathrm{A} & \mathrm{C} & \mathrm{A} & \mathrm{A} & \mathrm{B} \\ \hline \text { 3rd place } & \mathrm{D} & \mathrm{A} & \mathrm{C} & \mathrm{A} & \mathrm{C} & \mathrm{C} & \mathrm{A} & \mathrm{D} & \mathrm{C} & \mathrm{C} \\ \hline \text { 4th place } & \mathrm{C} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{C} & \mathrm{D} & \mathrm{D} \end{array}$ $\text { Write a voter profile for these ballots. }$

A) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 2 & \mathrm{~A}>\mathrm{B}>\mathrm{D}>\mathrm{C} \\ 2 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ B) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~A}>\mathrm{B}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ C) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~A}>\mathrm{C}>\mathrm{B}>\mathrm{D} \\ 3 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ D) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 2 & \mathrm{~A}>\mathrm{C}>\mathrm{B}>\mathrm{D} \\ 2 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ A) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 2 & \mathrm{~A}>\mathrm{B}>\mathrm{D}>\mathrm{C} \\ 2 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ B) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~A}>\mathrm{B}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ C) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~A}>\mathrm{C}>\mathrm{B}>\mathrm{D} \\ 3 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$ D) $\begin{array}{c|l} \text { Number of Votes } & \text { Ranking } \\ \hline 2 & \mathrm{~A}>\mathrm{B}>\mathrm{C}>\mathrm{D} \\ 2 & \mathrm{~A}>\mathrm{C}>\mathrm{B}>\mathrm{D} \\ 2 & \mathrm{~B}>\mathrm{A}>\mathrm{C}>\mathrm{D} \\ 1 & \mathrm{~B}>\mathrm{A}>\mathrm{D}>\mathrm{C} \\ 3 & \mathrm{~B}>\mathrm{C}>\mathrm{A}>\mathrm{D} \end{array}$

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion For the voter profile? $\begin{array}{c|l} \text { Number of Voters } & {\text { Ranking }} \\ \hline 3 & \mathrm{a}>\mathrm{c}>\mathrm{b}>\mathrm{d} \\ 7 & \mathrm{c}>\mathrm{b}>\mathrm{a}>\mathrm{d} \\ 5 & \mathrm{a}>\mathrm{b}>\mathrm{c}>\mathrm{d} \\ 8 & \mathrm{a}>\mathrm{c}>\mathrm{d}>\mathrm{b} \\ 11 & \mathrm{c}>\mathrm{d}>\mathrm{a}>\mathrm{b} \\ 4 & \mathrm{~b}>\mathrm{d}>\mathrm{a}>\mathrm{c} \\ 6 & \mathrm{~d}>\mathrm{b}>\mathrm{a}>\mathrm{c} \\ 9 & \mathrm{c}>\mathrm{b}>\mathrm{d}>\mathrm{a} \\ 7 & \mathrm{c}>\mathrm{a}>\mathrm{d}>\mathrm{b} \end{array}$

A) a, a; no B) a, c; yes C) c, c; no D) c, a; yes A) a, a; no B) a, c; yes C) c, c; no D) c, a; yes

Exam 16: Voting and Apportionment

Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> -Construct a voter profile for 41 voters and 4 candidates that has a Condorcet Candidate that fails to be elected by both the Borda Method and the Hare Method.

(Essay)

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Question 61

Number of Voters Ranking 21 >>>> 20 >>>> -Given that there are 30 voters and 3 candidates, complete the following voter profile so that the Borda Method violates the Majority Criterion. Number of Voters Ranking 16 >> ? >> ? >>

(Essay)

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Question 62

Suppose 8 candidates a, b, c, d, e, f, g, h are involved in a Pairwise Comparison Method election. The number of pairwise comparisons each candidate wins is shown in the table. Candidate Number of Pairwise Comparisons Won 4 5 ? 3 4 2 1 4 How many comparisons does candidate c win?

(Multiple Choice)

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Question 63

State Population 210 165 160 175 190 -Create a population profile for 5 states with a total population of 8,000,000 for which the Hamilton Method and Webster Method apportionments of 250 legislative seats are the same, and the Jefferson Method apportionment is different.

(Essay)

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Question 64

For the voter profile given below, who (or what)is selected if a runoff election is held between the candidates that rank second and third; and the winner of that election faces the candidate with the Most first place votes, to decide the final outcome of the election? Number of Voters Ranking 2 >> 3 >> 3 >> 1 >> 5 >> 1 >>

(Multiple Choice)

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Question 65

Which voting method always satisfies the monotonicity criterion?

(Multiple Choice)

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Question 66

Solve the problem. -The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with Candidates Clark (C), Jones (J), and Smith (S). Number of Voters Ranking 7 >> 10 >> 4 >> 8 >> Determine the winner using the pairwise comparison method and state whether the winner Received a majority of votes.

(Multiple Choice)

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Question 67

Which apportionment method never violates the quota rule?

(Multiple Choice)

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Question 68

Central City Community College is made up of four divisions: Liberal Arts, Sciences, Business, and Humanities. The faculty senate has 50 senators to be apportioned based on the faculty of each Division as shown below. Division Liberal Arts Sciences Business Humanities Faculty 130 175 46 49 Use the Adams Method to apportion the senators.

(Multiple Choice)

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Question 69

Find the quantity requested. Round to the nearest whole number. -A total of 156 seats were apportioned to 19 states in a certain country which has a total population of 8,977,808. Find the average number of people represented per seat in this country.

(Multiple Choice)

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Question 70

Which apportionment method may produce the population paradox?

(Multiple Choice)

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Question 71

Which fairness criterion is always satisfied by the Hare method?

(Multiple Choice)

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Question 72

In the following voter profile, use the Pairwise Comparison Method to determine the winner. If candidate c drops out, and a second election is held, who is the winner if the Pairwise Comparison Method is again used? Does the Pairwise Comparison Method violate the Independence of Irrelevant Alternatives Criterion in this election process? Number of Voters Ranking 7 >> 10 >> 4 >> 8 >>

(Multiple Choice)

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Question 73

Solve the problem. -The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with Candidates Clark (C), Jones (J), and Smith (S). Number of Voters Ranking 7 >> 10 >> 4 >> 8 >> Determine the winner using the plurality method and state whether the winner received a majority Of votes.

(Multiple Choice)

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Question 74

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to The following voter profile. Number of Votes Ranking 5 >>> 7 >>> 4 >>> 7 >>> 5 >>> 6 >>> 8 >>> Determine the winner using the pairwise comparison method.

(Multiple Choice)

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Question 75

Write a voter profile for the voting situation. -Ten voters are asked to rank 4 brands of cell phones: A, B, C, and D. The ten voters turn in the following ballots showing their preferences in order: Voter \#1 \#2 \#3 \#4 \#5 \#6 \#7 \#8 \#9 \#10 1st place 2nd place 3rd place 4th place $\text { Write a voter profile for these ballots. }$

(Multiple Choice)

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Question 76

State Population 131 140 304 178 197 -If the Hamilton Method is used to apportion legislative seats to four states with the populations given in the table, then the Alabama Paradox occurs when the number of seats being apportioned Increases from 204 to 205. State Population 3462 7470 4265 5300 In which state does the paradox show up?

(Multiple Choice)

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Question 77

Which apportionment paradox may be produced by the Adams' method but not the Webster method?

(Multiple Choice)

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Question 78

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion For the voter profile? Number of Voters Ranking 3 >>> 7 >>> 5 >>> 8 >>> 11 >>> 4 >>> 6 >>> 9 >>> 7 >>>

(Multiple Choice)

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Question 79

Which voting method always satisfies the Condorcet criterion?

(Multiple Choice)

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Question 80

Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> -Construct a voter profile for 41 voters and 4 candidates that has a Condorcet Candidate that fails to be elected by both the Borda Method and the Hare Method.

Number of Voters Ranking 21 >>>> 20 >>>> -Given that there are 30 voters and 3 candidates, complete the following voter profile so that the Borda Method violates the Majority Criterion. Number of Voters Ranking 16 >> ? >> ? >>

Suppose 8 candidates a, b, c, d, e, f, g, h are involved in a Pairwise Comparison Method election. The number of pairwise comparisons each candidate wins is shown in the table. Candidate Number of Pairwise Comparisons Won 4 5 ? 3 4 2 1 4 How many comparisons does candidate c win?

State Population 210 165 160 175 190 -Create a population profile for 5 states with a total population of 8,000,000 for which the Hamilton Method and Webster Method apportionments of 250 legislative seats are the same, and the Jefferson Method apportionment is different.

Which voting method always satisfies the monotonicity criterion?

Which apportionment method never violates the quota rule?

Find the quantity requested. Round to the nearest whole number. -A total of 156 seats were apportioned to 19 states in a certain country which has a total population of 8,977,808. Find the average number of people represented per seat in this country.

Which apportionment method may produce the population paradox?

Which fairness criterion is always satisfied by the Hare method?

Which apportionment paradox may be produced by the Adams' method but not the Webster method?

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion For the voter profile? Number of Voters Ranking 3 >>> 7 >>> 5 >>> 8 >>> 11 >>> 4 >>> 6 >>> 9 >>> 7 >>>

Which voting method always satisfies the Condorcet criterion?

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