$\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 16 & \mathrm {~b} > \mathrm { d } > \mathrm { c } > \mathrm { a } \\ 9 & \mathrm { c } > \mathrm { d } > \mathrm { b } > \mathrm { a } \\ 5 & \mathrm { c } > \mathrm { d } > \mathrm { a } > \mathrm { b } \end{array}$ -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 Plurality Method.

To construct a voter profile for 41 vote

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). $\begin{array}{l} \text { Number of Votes Ranking }\\ \begin{array} { l | l } \hline 3 & \mathrm {~A} > \mathrm { C } > \mathrm { B } > \mathrm { D } \\ 7 & \mathrm {~B} > \mathrm { A } > \mathrm { C } > \mathrm { D } \\ 5 & \mathrm {~A} > \mathrm { B } > \mathrm { C } > \mathrm { D } \\ 8 & \mathrm {~A} > \mathrm { C } > \mathrm { D } > \mathrm { B } \\ 7 & \mathrm { C } > \mathrm { D } > \mathrm { A } > \mathrm { B } \\ 4 & \mathrm {~B} > \mathrm { D } > \mathrm { A } > \mathrm { C } \\ 6 & \mathrm { D } > \mathrm { B } > \mathrm { A } > \mathrm { C } \\ 6 & \mathrm { C } > \mathrm { B } > \mathrm { D } > \mathrm { A } \\ 4 & \mathrm { C } > \mathrm { A } > \mathrm { D } > \mathrm { B } \end{array} \end{array}$ Determine the winner using the Borda count method and state whether the winner received a Majority of votes.

A) Downs; no B) Abbott; no C) Blake; no D) Cleary; no A) Downs; no B) Abbott; no C) Blake; no D) Cleary; no

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. $\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm {~b} > \mathrm { a } > \mathrm { c } > \mathrm { d } \\ 11 & \mathrm { c } > \mathrm { a } > \mathrm { d } > \mathrm { b } \\ 9 & \mathrm {~d} > \mathrm { c } > \mathrm { a } > \mathrm { b } \\ 5 & \mathrm { a } > \mathrm { d } > \mathrm { c } > \mathrm { b } \\ 1 & \mathrm { c } > \mathrm { d } > \mathrm { a } > \mathrm { b } \end{array}$ Use the Hare Method to determine the winner.

A) Carol B) Andy C) Denise D) Brett A) Carol B) Andy C) Denise D) Brett

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 Hare 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

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

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

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Hare Method. Does the Hare Method violate the Condorcet criterion For the voter profile? $\begin{array}{c|l} \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm{~b}>\mathrm{a}>\mathrm{c}>\mathrm{d} \\ 11 & \mathrm{c}>\mathrm{a}>\mathrm{d}>\mathrm{b} \\ 9 & \mathrm{~d}>\mathrm{c}>\mathrm{a}>\mathrm{b} \\ 5 & \mathrm{a}>\mathrm{d}>\mathrm{c}>\mathrm{b} \\ 1 & \mathrm{c}>\mathrm{d}>\mathrm{a}>\mathrm{b} \end{array}$

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

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

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

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 initially rank the four possible sites According to the following voter profile: \[\begin{array} { c | l } \text { Number of Voters } & { \text { Ranking } } \\ \hline 5 & \mathrm {~A} > \mathrm { H } > \mathrm { F } > \mathrm { S } \\ 7 & \mathrm { H } > \mathrm { S } > \mathrm { A } > \mathrm { F } \\ 4 & \mathrm {~F} > \mathrm { S } > \mathrm { H } > \mathrm { A } \\ 7 & \mathrm {~A} > \mathrm { F } > \mathrm { S } > \mathrm { H } \\ 5 & \mathrm {~A} > \mathrm { H } > \mathrm { S } > \mathrm { F } \\ 6 & \mathrm { H } > \mathrm { F } > \mathrm { S } > \mathrm { A } \\ 8 & \mathrm {~F} > \mathrm { S } > \mathrm { A } > \mathrm { H } \end{array}\] Use the Hare method to determine the winner of a preliminary nonbinding election based on the above voter profile. Suppose that after the preliminary election, the four voters who voted $\mathrm { F } > \mathrm { S } > \mathrm { H } > \mathrm { A }$, in that order, change their votes to $\mathrm { A } > \mathrm { F } > \mathrm { S } > \mathrm { H }$. The official vote is then held. If the Hare Method is used to determine the winner, who is the winner and is the Monotonicity Criterion violated?

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

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. $\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm {~b} > \mathrm { a } > \mathrm { c } > \mathrm { d } \\ 11 & \mathrm { c } > \mathrm { a } > \mathrm { d } > \mathrm { b } \\ 9 & \mathrm {~d} > \mathrm { c } > \mathrm { a } > \mathrm { b } \\ 5 & \mathrm { a } > \mathrm { d } > \mathrm { c } > \mathrm { b } \\ 1 & \mathrm { c } > \mathrm { d } > \mathrm { a } > \mathrm { b } \end{array}$ Use the Pairwise Comparison Method to determine the winner.

A) Denise B) Andy C) Brett D) Carol A) Denise B) Andy C) Brett D) Carol

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 preliminary Nonbinding ballots with candidates Clark (C), Jones (J), and Smith (S). $\begin{array}{c|l} \text { Number of Voters } & \text { Ranking } \\ \hline 7 & \mathrm{~J}>\mathrm{S}>\mathrm{C} \\ 10 & \mathrm{~S}>\mathrm{J}>\mathrm{C} \\ 4 & \mathrm{~J}>\mathrm{C}>\mathrm{S} \\ 8 & \mathrm{C}>\mathrm{S}>\mathrm{J} \end{array}$ A second election is then held with the following voter profile: $\begin{array}{c|l} \text { Number of Voters } & \text { Ranking } \\ \hline 7 & \mathrm{~J}>\mathrm{S}>\mathrm{C} \\ 11 & \mathrm{~S}>\mathrm{J}>\mathrm{C} \\ 3 & \mathrm{~J}>\mathrm{C}>\mathrm{S} \\ 8 & \mathrm{C}>\mathrm{S}>\mathrm{I} \end{array}$ Use the Pairwise Comparison Method to determine the winner, and determine if the Monotonicity Criterion is violated in this selection process.

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

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. $\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\ \hline 15 & \mathrm {~b} > \mathrm { a } > \mathrm { c } > \mathrm { d } \\ 11 & \mathrm { c } > \mathrm { a } > \mathrm { d } > \mathrm { b } \\ 9 & \mathrm {~d} > \mathrm { c } > \mathrm { a } > \mathrm { b } \\ 5 & \mathrm { a } > \mathrm { d } > \mathrm { c } > \mathrm { b } \\ 1 & \mathrm { c } > \mathrm { d } > \mathrm { a } > \mathrm { b } \end{array}$ Use the Plurality Method to determine the winner.

A) Andy B) Carol C) Brett D) Denise A) Andy B) Carol C) Brett D) Denise

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

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

Exam 16: Voting and Apportionment

Number of Voters Ranking 16 >>> 9 >>> 5 >>> -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 Plurality Method.

(Short Answer)

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Question 21

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

(Multiple Choice)

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Question 22

Find the Hill-Huntington cutoff point for rounding up a modified quota of 5.671.

(Multiple Choice)

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Question 23

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> Use the Hare Method to determine the winner.

(Multiple Choice)

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Question 24

Solve the problem. -Create a population profile for 5 states with a total population of 900 for which the Hamilton Method, Jefferson Method, and Webster Method apportionments of 50 legislative seats all agree.

(Essay)

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Question 25

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

(Multiple Choice)

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Question 26

State Population 131 140 304 178 197 -If the Hamilton Method is used to apportion legislative seats to three original states and a new state with the populations given in the table, then one of the states loses a seat to another if the original Number of seats being apportioned is 60. State Original State a Original State b Original State c New State d Population 62,700 230,700 606,600 78,000 Which state loses a seat to which state, and does the New States paradox occur?

(Multiple Choice)

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Question 27

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

(Multiple Choice)

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Question 28

In the following voter profile, use the Borda Method to determine the winner. If candidate d drops out, and a second election is held, who is the winner if the Borda Method is again used? Does the Borda Method violate the Independence of Irrelevant Alternatives Criterion in this election Process? Number of Voters Ranking 3 a>c>b>d 7 b>a>c>d 5 a>b>c>d 8 a>c>d>b 7 c>d>a>b 4 b>d>a>c 6 d>b>a>c 6 c>b>d>a 4 c>a>d>b

(Multiple Choice)

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Question 29

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 7 a>b>c 10 b>a>c 12 a>c>b 8 c>b>a

(Multiple Choice)

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Question 30

State Population 131 140 304 178 197 -If the Hamilton Method is used to apportion legislative seats to two original states and a new state with the populations given in the table (in thousands), then one of the states loses a seat to another If the original number of seats being apportioned is 48. State Original State a Original State b New State c Population 844 3956 724 Which state loses a seat to which state, and does the New States paradox occur?

(Multiple Choice)

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Question 31

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Hare Method. Does the Hare Method violate the Condorcet criterion For the voter profile? Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>>

(Multiple Choice)

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Question 32

Solve the problem. -A small country consists of 7 provinces with the following populations: Province A B C D E F G 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 Hamilton's method.

(Multiple Choice)

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Question 33

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 initially rank the four possible sites According to the following voter profile: Number of Voters Ranking 5 >>> 7 >>> 4 >>> 7 >>> 5 >>> 6 >>> 8 >>> Use the Hare method to determine the winner of a preliminary nonbinding election based on the above voter profile. Suppose that after the preliminary election, the four voters who voted $\mathrm { F } > \mathrm { S } > \mathrm { H } > \mathrm { A }$ , in that order, change their votes to $\mathrm { A } > \mathrm { F } > \mathrm { S } > \mathrm { H }$ . The official vote is then held. If the Hare Method is used to determine the winner, who is the winner and is the Monotonicity Criterion violated?

(Multiple Choice)

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Question 34

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> Use the Pairwise Comparison Method to determine the winner.

(Multiple Choice)

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Question 35

Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> -Suppose that one of the losing candidates is to be deleted from the given voter profile. Which choices would result in the plurality method violating the independence of irrelevant alternatives Criterion in a second election? Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>>

(Multiple Choice)

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Question 36

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 preliminary Nonbinding ballots with candidates Clark (C), Jones (J), and Smith (S). Number of Voters Ranking 7 >> 10 >> 4 >> 8 >> A second election is then held with the following voter profile: Number of Voters Ranking 7 >> 11 >> 3 >> 8 >> Use the Pairwise Comparison Method to determine the winner, and determine if the Monotonicity Criterion is violated in this selection process.

(Multiple Choice)

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(41)

Question 37

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> Use the Plurality Method to determine the winner.

(Multiple Choice)

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(31)

Question 38

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 Adams' method.

(Multiple Choice)

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(32)

Question 39

Solve the problem. -West End school district is purchasing 50 new computers to be distributed among the five schools in the district. The computers will be apportioned based on the school population of each school as Given in the table below. Use the Hamilton Method to apportion the computers. School Population 210 165 160 175 190

(Multiple Choice)

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Question 40

Number of Voters Ranking 16 >>> 9 >>> 5 >>> -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 Plurality Method.

Find the Hill-Huntington cutoff point for rounding up a modified quota of 5.671.

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> Use the Hare Method to determine the winner.

Solve the problem. -Create a population profile for 5 states with a total population of 900 for which the Hamilton Method, Jefferson Method, and Webster Method apportionments of 50 legislative seats all agree.

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

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 7 a>b>c 10 b>a>c 12 a>c>b 8 c>b>a

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Hare Method. Does the Hare Method violate the Condorcet criterion For the voter profile? Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>>

Solve the problem. -Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Number of Voters Ranking 15 >>> 11 >>> 9 >>> 5 >>> 1 >>> Use the Plurality Method to determine the winner.

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 Adams' method.

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