Deck 13: Graph Theory

Question 1

(a) Using the pairwise comparison voting method, determine the winner. (b) If candidate S is eliminated and the votes are recounted, does this election violate the irrelevant alternative criterion?

Answer 1

(a) W
(b) No. W still wins.

Question 2

Answer 2

(a) X
(b) No, the winner is still X.

Question 3

(a) How many people voted? (b) Determine the winner using the plurality method.

Answer 3

(a) 30
(b) S

Question 4

Is the monotonicity criterion violated? Explain your answer.

Question 5

Does the election violate the majority criterion? Explain your answer.

Question 6

(a) How many people voted? (b) How many people voted for the candidates in the order of preference BCA? (c) How many people voted for candidate B for their first choice?

Question 7

Question 8

Question 9

(a) Using the pairwise comparison voting method, determine the winner. (b) If candidate X drops out and the votes are recounted, does this election violate the irrelevant alternative criterion?

Question 10

Question 11

(a) Using the Borda count method of voting, determine the winner. (b) Does the election violate the majority criterion?

Question 12

(a) How many people voted? (b) How many people voted for the candidates in the order of preference ZYX? (c) How many people voted for candidate X for their first choice?

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

If there are 7 candidates in an election, how many pairwise comparisons need to be made in order to determine a winner?

Question 19

Question 20

(a) Using the Borda count method of voting, determine the winner. (b) Does the election violate the majority criterion?

Question 21

The results of an election are summarized in the preference table below.

$\begin{array}{l|cccc}\text { Number of votes } & 5 & 4 & 6 & 3 \\\hline \text { First choice } & \text { B } & \text { A } & \text { D } & \text { C } \\\text { Second choice } & \text { C } & \text { C } & \text { A } & \text { B } \\\text { Third choice } & \text { A } & \text { B } & \text { B } & \text { A } \\\text { Fourth choice } & \text { D } & \text { D } & \text { C } & \text { D }\end{array}$

Using the plurality-with-elimination method of voting, determine the winner.

A) D
B) A
C) C
D) B

Question 22

The results of an election are summarized in the following preference table. If the plurality method is used, determine whether the head-to-head comparison criterion has been violated and explain why.

$\begin{array}{l|lllll}\text { Number of votes } & 5 & 12 & 7 & 14 & 10 \\\hline \text { First choice } & \mathrm{Z} & \mathrm{Z} & \mathrm{X} & \mathrm{X} & \mathrm{Y} \\\text { Second choice } & \mathrm{X} & \mathrm{Y} & \mathrm{Y} & \mathrm{Z} & \mathrm{Z} \\\text { Third choice } & \mathrm{Y} & \mathrm{X} & \mathrm{Z} & \mathrm{Y} & \mathrm{X}\end{array}$

A) No, because X is the plurality winner.
B) Yes, because X does not win over Z in a head-to-head comparison.
C) Yes, because X does not win over Y in a head-to-head comparison.
D) No, because X wins over Y and Z in a head-to-head comparison.

Question 23

Question 24

Use a preference table for the following preference ballots to determine the winner of the election using the plurality method.

$\begin{array}{llllllllllllll}\text { B } & \text { B } & \text { C } & \text { B } & \text { B } & \text { C } & \text { B } & \text { C } & \text { B } & \text { B } & \text { B } & \text { B } & \text { B } & \text { B } \\\text { C } & \text { A } & \text { B } & \text { C } & \text { C } & \text { A } & \text { A } & \text { B } & \text { C } & \text { A } & \text { A } & \text { A } & \text { C } & \text { A } \\\text { A } & \text { C } & \text { A } & \text { A } & \text { A } & \text { B } & \text { C } & \text { A } & \text { A } & \text { C } & \text { C } & \text { C } & \text { A } & \text { C }\end{array}$

$\begin{array}{llllllll}\mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{C} & \mathrm{C} & \mathrm{C} \\\mathrm{C} & \mathrm{B} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{B} & \mathrm{B} & \mathrm{A} \\\mathrm{A} & \mathrm{A} & \mathrm{C} & \mathrm{B} & \mathrm{C} & \mathrm{A} & \mathrm{A} & \mathrm{B}\end{array}$

A) Candidate B wins.
B) Candidates A and B tie.
C) Candidate A wins.
D) Candidate C wins.

Question 25

(a) Determine the winner using the Borda method of voting. (b) Is the winner the same as the one determined by the plurality method?

Question 26

Question 27

Question 28

Question 29

The results of an election are summarized in the following preference table. Determine the winner using the plurality method.
$\begin{array}{l|cccc}\text { Number of votes } & 3 & 7 & 7 & 8 \\\hline \text { First choice } & \mathrm{K} & \mathrm{L} & \mathrm{K} & \text { J } \\\text { Second choice } & \text { L } & \text { K } & \text { J } & \text { K } \\\text { Third choice } & \text { J } & \text { J } & \text { L } & \text { L }\end{array}$

A) J
B) Tie between K and L
C) L
D) K

Question 30

Question 31

(a) Determine the winner using the Borda method of voting. (b) Is the winner the same as the one determined by the plurality method?

Question 32

A hiring board must select a new teacher. They vote using the approval method. The results are shown.


Which applicant was selected?

A) Phillips
B) James
C) Hanson
D) Boyd

Question 33

Question 34

The results of an election are summarized in the preference table below.
$\begin{array}{l|lllll}\text { Number of votes } & 4 & 2 & 5 & 4 & 1 \\\hline \text { First choice } & \text { Z } & \text { X } & \text { W } & \text { Y } & \text { X } \\\text { Second choice } & \text { W } & \text { W } & \text { Y } & \text { Z } & \text { Z } \\\text { Third choice } & \text { Y } & \text { Y } & \text { Z } & \text { X } & \text { W } \\\text { Fourth choice } & \text { X } & \text { Z } & \text { X } & \text { W } & \text { Y }\end{array}$

Using the plurality-with-elimination method of voting, determine the winner.

A) Y
B) X
C) W
D) Z

Question 35

The results of an election are summarized in the preference table below.

$\begin{array}{l|llll}\text { Number of votes } & 10 & 8 & 12 & 14 \\\hline \text { First choice } & \text { B } & \text { A } & \text { B } & \text { C } \\\text { Second choice } & \text { A } & \text { C } & \text { C } & \text { A } \\\text { Third choice } & \text { C } & \text { B } & \text { A } & \text { B }\end{array}$

Using the pairwise comparison voting method, determine the winner.

A) C
B) A
C) B

Question 36

A county library has three branches. The number of books at each branch of the library is shown below. The head librarian purchased 49 new books and apportioned them to the libraries using Hamilton's method. When the book order arrived the librarian noticed there was an extra book added as a "bonus" and she reapportioned the books with the extra book added. Determine if the reapportionment resulted in the Alabama paradox

$\begin{array}{l|cccr}\text { Branch } & \text { A } & \text { B } & \text { C } & \text { Total } \\\hline \text { Books } & 1,598 & 1,379 & 388 & 3,365\end{array}$

A) Alabama paradox did not occur
B) Alabama paradox occurred

Question 37

A city with three districts has an increase in population as indicated in the table, Using Hamilton's method, apportion the 30 seats between the three districts both before and after the population increase. Look at the percent increase in population for each district to determine if the population paradox occurred.
$\begin{array}{l|cccc}\text { District } & \text { A } & \text { B } & \text { C } & \text { Total } \\\hline \text { Original population (in thousands) } & 954 & 545 & 2,334 & 3,833 \\\text { New population (in thousands) } & 1,205 & 696 & 2,677 & 4,578\end{array}$

A) population paradox occurred
B) population paradox did not occur

Question 38

A) No. In a head-to-head comparison Z won over C and W.
B) Yes. In a head-to-head comparison Z won over W but lost to C.
C) No. Z is the plurality winner.
D) Yes. In a head-to-head comparison Z won over C but lost to W.

Question 39

Question 40

Question 41

The preference table below summarizes the results of an election.
$\begin{array}{l|cccc}\text { Number of votes } & 5 & 5 & 6 & 7 \\\hline \text { First choice } & \text { F } & \text { L } & \text { Q } & \text { M } \\\text { Second choice } & \text { L } & \text { M } & \text { F } & \text { Q } \\\text { Third choice } & \text { Q } & \text { F } & \text { L } & \text { L } \\\text { Fourth choice } & \text { M } & \text { Q } & \text { M } & \text { F }\end{array}$

Using the Borda count method of voting, determine the winner.

A) Q
B) F
C) M
D) L

Question 42

The math department is voting for a new department chairperson. The candidates are Jenkins (J), Peterson (P), and Thomas (T). The results of the election are summarized in the preference table below.
$\begin{array}{l|lll}\text { Number of votes } & 5 & 3 & 4 \\\hline \text { First choice } & \text { T } & \text { J } & \text { P } \\\text { Second choice } & \text { J } & \text { P } & \text { T } \\\text { Third choice } & \text { P } & \text { T } & \text { J }\end{array}$

Using the plurality-with-elimination method of voting, determine the winner.

A) Peterson (P)
B) Thomas (T)
C) Jenkins (J)

Question 43

Question 44

Question 45

Question 46

A club voted on what time to have their next meeting $( 12 : 00$ p.m., $1 : 00$ p.m., 5:00 p.m., or 6:00 p.m.). The election results are summarized in the preference table below.
$\begin{array}{l|rrrr}\text { Number of votes } & 4 & 15 & 6 & 10 \\\hline \text { First choice } & 5 & 6 & 6 & 12 \\\text { Second choice } & 6 & 5 & 12 & 1 \\\text { Third choice } & 1 & 12 & 5 & 6 \\\text { Fourth choice } & 12 & 1 & 1 & 5\end{array}$

Using the Borda count method of voting, determine the winner.

A) 6:00 p.m.
B) 12:00 p.m.
C) 5:00 p.m.
D) 1:00 p.m.

Question 47

The results of an election are summarized in the preference table below.
$\begin{array}{l|lllll}\text { Number of votes } & 5 & 12 & 7 & 14 & 10 \\\hline \text { First choice } & \mathrm{Z} & \mathrm{Z} & \mathrm{X} & \mathrm{X} & \mathrm{Y} \\\text { Second choice } & \mathrm{X} & \mathrm{Y} & \mathrm{Y} & \mathrm{Z} & \mathrm{Z} \\\text { Third choice } & \mathrm{Y} & \mathrm{X} & \mathrm{Z} & \mathrm{Y} & \mathrm{X}\end{array}$

Using the plurality-with-elimination method of voting, determine the winner.

A) X
B) Z
C) Y

Question 48

Question 49

A committee must select one of its members to serve as president. They vote using the approval method. The results are shown.
$\begin{array}{l|cccc}\text { Number of votes } & 7 & 2 & 8 & 1 \\\hline \text { Morgan } & / & / & & / \\\text { Green } & & / & & / \\\text { Robinson } & & / & / & / \\\text { Park } & / & & / & /\end{array}$

Who was selected to be president of the committee?

A) Robinson
B) Green
C) Morgan
D) Park

Question 50

Question 51

A tenure board must select one teacher to get tenure. They vote using the approval method. The results are shown.
$\begin{array}{l|lllll}\text { Number of votes } & 5 & 3 & 3 & 4 & 4 \\\hline \text { Stewart } & & & / & / & / \\\text { Davis } & / & / & & & \\\text { Wong } & / & & & / & / \\\text { Luby } & & / & / & & /\end{array}$

Which teacher was selected?

A) Stewart
B) Wong
C) Luby
D) Davis

Question 52

A class of fifth graders voted on their favorite flavor of ice cream, vanilla $( \mathrm { V } )$ , chocolate $( \mathrm { C } )$ , or strawberry $( \mathrm { S } )$ . The election results are summarized in the preference table below.
$\begin{array}{l|cccc}\text { Number of votes } & 8 & 12 & 5 & 2 \\\hline \text { First choice } & \mathrm{C} & \mathrm{V} & \mathrm{C} & \mathrm{V} \\\text { Second choice } & \mathrm{V} & \mathrm{C} & \mathrm{S} & \mathrm{S} \\\text { Third choice } & \mathrm{S} & \mathrm{S} & \mathrm{V} & \mathrm{C}\end{array}$

Using the Borda count method of voting, determine the winner.

A) Vanilla (V)
B) Strawberry (S)
C) Chocolate (C)

Question 53

Question 54

The results of an election are summarized in the preference table below.
$\begin{array}{l|ccccc}\text { Number of votes } & 8 & 10 & 2 & 4 & 3 \\\hline \text { First choice } & \text { B } & \text { A } & \text { A } & \text { C } & \text { B } \\\text { Second choice } & \text { A } & \text { B } & \text { C } & \text { A } & \text { C } \\\text { Third choice } & \text { C } & \text { C } & \text { D } & \text { B } & \text { A } \\\text { Fourth choice } & \text { D } & \text { D } & \text { B } & \text { D } & \text { D }\end{array}$

Using the plurality-with-elimination method of voting, determine the winner.

A) A
B) C
C) B
D) D

Question 55

Question 56

The math department is voting for a new department chairperson. The candidates are Jenkins (J), Peterson (P), and Thomas (T). The results of the election are summarized in the preference table below.
$\begin{array}{l|lll}\text { Number of votes } & 5 & 3 & 4 \\\hline \text { First choice } & \text { T } & \text { J } & \text { P } \\\text { Second choice } & \text { J } & \text { P } & \text { T } \\\text { Third choice } & \text { P } & \text { T } & \text { J }\end{array}$

The winner is determined by the plurality-with-elimination method. Suppose the election is declared invalid, and that a reelection is held. If the voters in column 2 change their vote from JPT to PJT, is the monotonicity criterion violated?

A) Yes
B) No

Question 57

If there are 6 candidates in an election, how many pairwise comparisons need to be made in order to determine a winner?

A) 6
B) 30
C) 21
D) 15

Question 58

The results of an election are summarized in the preference table below.
$\begin{array}{l|lcccc}\text { Number of votes } & 5 & 12 & 7 & 14 & 10 \\\hline \text { First choice } & \mathrm{Z} & \mathrm{Z} & \mathrm{X} & \mathrm{X} & \mathrm{Y} \\\text { Second choice } & \mathrm{X} & \mathrm{Y} & \mathrm{Y} & \mathrm{Z} & \mathrm{Z} \\\text { Third choice } & \mathrm{Y} & \mathrm{X} & \mathrm{Z} & \mathrm{Y} & \mathrm{X}\end{array}$

Suppose the election is declared invalid, and a reelection is held. If the people who voted $\mathrm { XYZ }$ in the original election change their vote to $\mathrm { ZXY }$ , is the monotonicity criterion violated?

A) No
B) Yes