For the voter profile given below, who (or what)is selected if a second runoff election is held between the two candidates with the most first place votes? $\begin{array}{c|l} \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm{~b}>\mathrm{a}>\mathrm{c}>\mathrm{d} \\ 11 & \mathrm{c}>\mathrm{a}>\mathrm{d}>\mathrm{b} \\ 9 & \mathrm{~d}>\mathrm{c}>\mathrm{a}>\mathrm{b} \\ 5 & \mathrm{a}>\mathrm{d}>\mathrm{c}>\mathrm{b} \\ 1 & \mathrm{c}>\mathrm{d}>\mathrm{a}>\mathrm{b} \end{array}$

A) $\mathrm { c }$ B) $b$ C) $\mathrm { d }$ D) $a$ A) $\mathrm { c }$ B) $b$ C) $\mathrm { d }$ D) $a$

Solve the problem. -A small country consists of 7 provinces with the following populations: $\begin{array}{lccccccc} \hline \text { Province } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} \\ \hline \text { Population } & 25,312 & 19,734 & 33,407 & 29,591 & 13,288 & 22,751 & 31,992 \\ \hline \end{array}$ There are 300 federal judges to be apportioned according to the population of each province. Find the apportionment for province $G$ using Jefferson's method.

A) 54 B) 53 C) 55 D) 56 A) 54 B) 53 C) 55 D) 56

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? $\begin{array}{c|l} \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm{~b}>\mathrm{a}>\mathrm{c}>\mathrm{d} \\ 11 & \mathrm{c}>\mathrm{a}>\mathrm{d}>\mathrm{b} \\ 9 & \mathrm{~d}>\mathrm{c}>\mathrm{a}>\mathrm{b} \\ 5 & \mathrm{a}>\mathrm{d}>\mathrm{c}>\mathrm{b} \\ 1 & \mathrm{c}>\mathrm{d}>\mathrm{a}>\mathrm{b} \end{array}$

A) $\mathrm { C }$ B) $b$ C) $a$ D) $\mathrm { d }$ A) $\mathrm { C }$ B) $b$ C) $a$ D) $\mathrm { d }$

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 Borda count method and state whether the winner received a Majority of votes.

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

Solve the problem. -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 Hare method and state whether the winner received a majority of Votes.

A) Hawaii; yes B) Alaska; yes C) Alaska; no D) Hawaii; no A) Hawaii; yes B) Alaska; yes C) Alaska; no D) Hawaii; no

Solve the problem. -A condominium association is holding an election for president of the board of directors. Each member ranks the candidates from first to third. The voter profile below shows the results of the Ballots with candidates Abbott(A), Blake (B), Cleary (C), and Downs (D). $\begin{array}{l} \text { Number of Votes Ranking }\\ \begin{array} { c | l } 3 & \mathrm {~A} > \mathrm { C } > \mathrm { B } > \mathrm { D } \\ 7 & \mathrm {~B} > \mathrm { A } > \mathrm { C } > \mathrm { D } \\ 5 & \mathrm {~A} > \mathrm { B } > \mathrm { C } > \mathrm { D } \\ 8 & \mathrm {~A} > \mathrm { C } > \mathrm { D } > \mathrm { B } \\ 7 & \mathrm { C } > \mathrm { D } > \mathrm { A } > \mathrm { B } \\ 4 & \mathrm {~B} > \mathrm { D } > \mathrm { A } > \mathrm { C } \\ 6 & \mathrm { D } > \mathrm { B } > \mathrm { A } > \mathrm { C } \\ 6 & \mathrm { C } > \mathrm { B } > \mathrm { D } > \mathrm { A } \\ 4 & \mathrm { C } > \mathrm { A } > \mathrm { D } > \mathrm { B } \end{array} \end{array}$ Determine the winner using the plurality method and state whether the winner received a majority Of votes.

A) Abbott; no B) Cleary; no C) Abbott; yes D) Cleary; yes A) Abbott; no B) Cleary; no C) Abbott; yes D) Cleary; yes

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. \[\begin{array} { l | l l l l l l l l l l } \text { Candidate } & \text { a } & \text { b } & \text { c } & \text { d } & \text { e } & \text { f } & \text { g } & \text { h } \\ \hline \text { Number of Pairwise Comparisons Won } & 4 & 5 & ? & 3 & 2 & 2 & 1 & 4 \end{array}\] Which candidate wins the election and how many pairwise points does the winner have?

A) c, with 7 pairwise points B) c, with 5 pairwise points C) c, with 8 pairwise points D) c, with 6 pairwise points A) c, with 7 pairwise points B) c, with 5 pairwise points C) c, with 8 pairwise points D) c, with 6 pairwise points

Solve the problem. -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 Borda count method and state whether the winner received a Majority of votes.

A) Hawaii; no B) Florida; no C) Alaska; no D) San Antonio; no A) Hawaii; no B) Florida; no C) Alaska; no D) San Antonio; no

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

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

Exam 16: Voting and Apportionment

For the voter profile given below, who (or what)is selected if a second runoff election is held between the two candidates with the most first place votes? Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>>

(Multiple Choice)

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Question 41

Solve the problem. -A small country consists of 7 provinces with the following populations: Province Population 25,312 19,734 33,407 29,591 13,288 22,751 31,992 There are 300 federal judges to be apportioned according to the population of each province. Find the apportionment for province $G$ using Jefferson's method.

(Multiple Choice)

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Question 42

State Population 131 140 304 178 197 -If the Jefferson Method is used to apportion 250 legislative seats to five states with the populations given in the table, then a violation of the Quota Rule occurs. State a b c d e Population 912,000 1,117,000 4,537,000 739,000 695,000 The apportionment for which state violates the Quota Rule?

(Multiple Choice)

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Question 43

Which voting method may violate the majority criterion?

(Multiple Choice)

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Question 44

State Population 131 140 304 178 197 -If the Jefferson Method is used to apportion 132 legislative seats to four states with the populations given in the table, then a violation of the Quota Rule occurs. State a b c d Population 17,180 7500 49,400 5820 The apportionment for which state violates the Quota Rule?

(Multiple Choice)

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Question 45

Which apportionment method may produce the new states paradox?

(Multiple Choice)

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Question 46

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 15 >>> 11 >>> 9 >>> 5 >>> 1 >>>

(Multiple Choice)

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Question 47

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 Borda count method and state whether the winner received a Majority of votes.

(Multiple Choice)

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Question 48

Which apportionment method may produce the Alabama paradox?

(Multiple Choice)

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Question 49

Solve the problem. -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 Hare method and state whether the winner received a majority of Votes.

(Multiple Choice)

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Question 50

Solve the problem. -A condominium association is holding an election for president of the board of directors. Each member ranks the candidates from first to third. The voter profile below shows the results of the Ballots with candidates Abbott(A), Blake (B), Cleary (C), and Downs (D). Number of Votes Ranking 3 >>> 7 >>> 5 >>> 8 >>> 7 >>> 4 >>> 6 >>> 6 >>> 4 >>> Determine the winner using the plurality method and state whether the winner received a majority Of votes.

(Multiple Choice)

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Question 51

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 a b c d e f g h Number of Pairwise Comparisons Won 4 5 ? 3 2 2 1 4 Which candidate wins the election and how many pairwise points does the winner have?

(Multiple Choice)

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Question 52

Eighteen voters are using the Borda method to select one of four candidates: a, b, c, or d. If Candidate a receives 35 Borda points, Candidate b receives 28 Borda points, and Candidate c Receives 20 Borda points, how many Borda points does Candidate d receive? Who wins the Borda Election?

(Multiple Choice)

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Question 53

Solve the problem. -A country has five states with populations as given in the table below and needs to apportion 250 seats in the legislature. Use the Webster Method to apportion the seats. State a b c d e Population 912,000 1,117,000 4,537,000 739,000 695,000

(Multiple Choice)

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Question 54

State Population 131 140 304 178 197 -If the Jefferson Method is used to apportion 131 legislative seats to six states with the populations given in the table, then a violation of the Quota Rule occurs. State Population 5672 8008 2400 6789 4972 20,000 The apportionment for which state violates the Quota Rule?

(Multiple Choice)

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Question 55

State Population 131 140 304 178 197 -If the Hamilton Method is used to apportion legislative seats to three states with the populations given in the table, then the Alabama Paradox occurs when the number of seats being apportioned Increases from 150 to 151. State a b c Population 740 6730 7530 In which state does the paradox show up?

(Multiple Choice)

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Question 56

Number of Voters Ranking 16 b>c>a 8 c>b>a 6 c>a>b -Given that there are 30 voters and 4 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 57

Solve the problem. -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 Borda count method and state whether the winner received a Majority of votes.

(Multiple Choice)

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Question 58

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

(Multiple Choice)

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Question 59

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Borda Method. Does the Borda Method violate the Condorcet Criterion for the voter profile? Number of Voters Ranking 7 a>b>c 10 b>a>c 4 a>c>b 8 c>b>a

(Multiple Choice)

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Question 60

For the voter profile given below, who (or what)is selected if a second runoff election is held between the two candidates with the most first place votes? Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>>

Which voting method may violate the majority criterion?

Which apportionment method may produce the new states paradox?

Which apportionment method may produce the Alabama paradox?

Eighteen voters are using the Borda method to select one of four candidates: a, b, c, or d. If Candidate a receives 35 Borda points, Candidate b receives 28 Borda points, and Candidate c Receives 20 Borda points, how many Borda points does Candidate d receive? Who wins the Borda Election?

Solve the problem. -A country has five states with populations as given in the table below and needs to apportion 250 seats in the legislature. Use the Webster Method to apportion the seats. State a b c d e Population 912,000 1,117,000 4,537,000 739,000 695,000

Number of Voters Ranking 16 b>c>a 8 c>b>a 6 c>a>b -Given that there are 30 voters and 4 candidates, complete the following voter profile so that the Borda Method violates the Majority Criterion. Number of Voters Ranking 16 >>> ? >>> ? >>>

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Borda Method. Does the Borda Method violate the Condorcet Criterion for the voter profile? Number of Voters Ranking 7 a>b>c 10 b>a>c 4 a>c>b 8 c>b>a

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