In Voting Among Three Candidates, the Outcomes Are Reported As Following Table

(a) If a person ranks B as their first choice, A as their second choice and C last, how would this be written?
( 1 )
(b) How many voters have this preference?
2

(a) If there are 40 voters and 5 candidates, how many total points would there be in a Borda count?
1
(b) If there are 70 voters and 4 candidates, how many total points would there be in a Borda count?
2
(c) If there are 600 voters and 3 candidates, how many total points would there be in a Borda count?
3

Use following formula to determine how many total points there would be in a Borda count with n voters and m candidates.

4. Twelve board members are voting on after-meeting activities, and they are asked to check any that they might like. The outcome of their choices is shown as following table. What is the outcome using approval voting?

1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
Snacks / X / X / X / X / X / X / X / X / X
Drinks / X / X / X / X / X / X / X
Travel talk / X / X / X
Speaker / X / X / X / X / X / X / X / X
5.

Twelve people serve on a board and are considering three alternatives: A, B, and C. The choices followed by vote is shown in following table. Determine the winner, if any, using a Borda count.
The winner is 1

(ABC) / (ACB) / (BAC) / (BCA) / (CAB) / (CBA)
2 / 4 / 2 / 1 / 2 / 1
6.

The South Davis Faculty Association is using the Hare method to vote for their collective bargaining representative. Their choices are the All Faculty Association (A), American Federation of Teachers (B), and California Teachers Association (C). The results of the preferences are shown in the following table.

(ABC) / (ACB) / (BAC) / (BCA) / (CAB) / (CBA)
13 / 1 / 3 / 7 / 3 / 20

(a) Which organization is selected for collective bargaining using the Hare method?

A B C

(b) Does the choice in part a violate the majority criterion?

Yes No

7. The Adobe School District is hiring a vice principal and has interviewed four candidates: Andrew (A), Bailey (B), Clark (C), and Davy (D). The results of the hiring are shown in the following table.

(ACDB) / (CBAD) / (BCDA) / (DBCA)
7 / 5 / 3 / 2

(a) Who is the winner using the plurality method?

A B B

(b) Suppose that Clark drops out of the running before the vote is taken. Who is the winner using the plurality method?

A B C

Yes No

The seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are shown in the following table.

(DABC) / (ACBD) / (BCAD) / (CBDA) / (CBAD)
80 / 45 / 30 / 10 / 50

(a) Who is the Condorcet candidate?

A B C D

(b) Is there a majority winner?

Yes No

A B C D

(d) Does (c) violate the Condorcet criterion?

Yes No

(e) Who wins the Borda count?

A B C D

(f) Does (e) violate the Condorcet criterion?

Yes No

(g) Who wins using the Hare method?

A B C D

(h) Does (g) violate the Condorcet criterion?

Yes No

(i) Who wins using the pairwise comparison method?

A B C D

(j) Does (i) violate the Condorcet criterion?

Yes No

A focus group of 33 people for ABC TV were asked to rank the government spending priorities of education (E), military spending (M), health care (H), immigration (I), and lowering taxes (T). The results of the preferences are shown in the following table.

(EIHTM) / (MIEHT) / (HMETI) / (TMEIH)
15 / 6 / 6 / 6

(a) Who is the winner using the pairwise comparison method?

E M H I T

(b) Who is the winner using a Borda count?

E M H I T

(c) Suppose that the losing issues of health care and lowering taxes are removed from the table. Now, who is the winner using the pairwise comparison method?

E M H I T

(d) Does (c) violate the irrelevant alternatives criterion?

Yes No

(e) Suppose that the losing issues of health care and lowering taxes are removed from the table. Now, who is the winner using the Borda count method?

E M H I T

(f) Does (e) violate the irrelevant alternatives criterion?

Yes No

(g) Suppose that the losing issues of health care, lowering taxes, and immigration are removed from the table. Now, who is the winner using the pairwise comparison method?

E M H I T

(h) Does the pairwise comparison method violate the irrelevant alternatives criterion?

Yes No

(i) Suppose that the losing issues of health care, lowering taxes, and immigration are removed from the table. Now, who is the winner using the Borda count method?

E M H I T

(j) Does the Borda count method violate the irrelevant alternatives criterion?

Yes No

10.

In 1993 the 101st International Olympic Committee met in Monaco to select the 2000 Winter Olympics site. The cities in the running were Beijing (B), Berlin (L), Istanbul (I), Manchester (M), and Sydney (S). Suppose we look at their voting preferences.

(BLIMS) / (LBSIM) / (IBLSM) / (MSBLI)
32 / 3 / 5 / 8
(LSBIM) / (SBLMI) / (IMSBL) / (MBSLI)
6 / 30 / 2 / 3

(a) Is there a majority winner?

yes no

(b) If not, which city wins the plurality vote?

B L I M S

(c) Find the results of the election using the Hare method. Just after the third vote, one of the committee members voting for Manchester was accused of cheating and was disqualified. Because of that scandal, one member admitted she was voting insincerely, and changed her vote from Manchester to Sydney. What is the result of using the Hare method?

B L I M S

(d) Do the results of parts (a) and (b) violate any of the fairness criteria?

yes no

(e) If the last answer positive, which fairness criterion is being violated?

None Condorcet Criterion Majority Criterion Monotonicity Criterion Irrelevant Alternatives Criterion

11.

Consider an election with three candidates with the following results:

(ACB) / (BAC) / (CBA) / (CAB)
6 / 13 / 10 / 6

(a) Is there a majority winner?

Yes No

(b) If not, who is the plurality winner?

A B C

(d) Who wins if they first eliminate the one with the most last-place votes and then have a runoff between the other two?

A B C

(e) Could the six voters with preference (CAB) change the outcome of the election in part (d) if they voted insincerely and pretended to have the preference (CBA)?

Yes No

Find the standard divisor for the given population and number of representative seats. Assume the population is equal to 120,000 and the number of seats is 6.
d = 1

Find the standard divisor (to two decimal places) for the given population and number of representative seats. Assume the population is equal to 9,440,000 and number of seats is 17.
d = 1

14.

Consider the following apportionment problem for College Town. Suppose each council member is to represent approximately 2,500 citizens. Use the apportionment plan requested in the problem, assuming there must be 10 representatives.

North: / 5,500
South: / 9,000
East: / 6,400
West: / 4,100

(a) Adams' plan.

North: / 1
South: / 2
East: / 3
West: / 4

(b) Jefferson's plan.

North: / 5
South: / 6
East: / 7
West: / 8

North: / 9
South: / 10
East: / 11
West: / 12

(d) Webster's plan.

North: / 13
South: / 14
East: / 15
West: / 16

(e) HH plan.

North: / 17
South: / 18
East: / 19
West: / 20

15. Consider the table.

State / A / B / C / D
Population / 685,000 / 347,000 / 160,000 / 95,000
Number of seats / 100

Use Adam's plan. Which state violates the quota rule?
1

16.

Consider the table.

State / A / B / C / D / E
Population / 1,100 / 1,100 / 1,550 / 4,590 / 2,010
Number of seats / 200

Use Jefferson's plan. Which state violates the quota rule?
1

17.

Use Hamilton's plan to apportion the new seats to the existing states. Which state illustrates the Alabama paradox? Assume that the populations are in thousands.
1

State / A / B / C / D / E
Population / 300 / 630 / 340 / 301 / 505
Number of seats / 50
18

Apportion the indicated number of representatives to two states, A, and B, using Hamilton's plan. Next, recalculate the apportionment using Hamilton's plan for the three states, C and the original states. Which state illustrates the new states paradox?
1

State / A / B / C
Population / 265,000 / 104,000 / 69,000
Number of original seats / 16
Number of additional seats / 2