Discrete Math

Chapter 1 Review: Election Theory

Use the preference schedules below for Questions 1-6 to determine the group preferences in each case. Show your work, giving the vote/point totals for each candidate where appropriate. You may wish to sketch the schedules again for some of the problems.

  1. Determine the group rankings, 1st-4th, for the candidates, using the plurality method. Remember to show the vote totals for each of the four candidates.

A: 20

C: 15

B: 12

D: 6

  1. Determine the group rankings using a 4-3-2-1 Borda count. Show your work.

A: 20(4) + 0(3) + 6(2) + 27(1) = 119  3rd

B: 12(4) + 41(3) + 0(2) + 0(1) = 171  1st

C: 15(4) + 0(3) + 32(2) + 6(1) = 130  2nd

D: 6(4) + 12(3) + 15(2) + 20(1) = 110  4th

  1. Determine the winner using a runoff.

A: 20 A: 26C wins.

C: 15C: 27

B: 12

D: 6

  1. Determine the winner using a sequential runoff. Show the results of each round.

A: 20A: 20 A: 20B wins.

C: 15C: 15B: 33

B: 12B: 18

D: 6

  1. Determine whether there is a Condorcet winner. Show a table.

A vs. B: 20-33A vs. C: 26-27A vs. D: 20-33

B vs. C: 38-15B vs. D: 47-6C vs. D: 35-18

A / B / C / D
A / L / L / L
B / W / W / W
C / W / L / W
D / W / L / L

B is undefeated in head-to-head competition, so it is a Condorcet winner.

  1. Using approval voting, assume that each candidate gets all of its first-place votes, and half of its second-place votes. What group rankings would result?

A: 20 + ½ (0) = 20

B: 12 + ½ (41) = 32.5 (or 33)

C: 15 + ½ (0) = 15

D: 6 + ½ (12) = 12

The group rankings would be B-A-C-D.

  1. Describe a paradox that would result if candidate B had to drop out of the race, and the runoff method were being used.

With candidate B removed, conduct the runoff again:

A: 20A: 20

C: 15D: 33

D: 18

This time the runoff is between A and D, and D wins. This is a paradox because D was not even part of the runoff originally. Removing B from the race should not have affected the outcome, because B did not win the runoff.

  1. Recall Arrow’s Conditions, summarized as follows:
  • Nondictatorship
  • Individual Sovereignty
  • Unanimity
  • Freedom from Irrelevant Alternatives
  • Uniqueness of the Group Ranking

Which of these conditions is violated by the scenario in Question 7? Explain.

Condition #4, freedom from irrelevant alternatives, was violated in Question 7. Candidate B did not originally win the runoff, so removing it

should have had no impact on the election.

  1. Consider a school where the Sophomore class is given 10 votes, the Junior class has 8 votes, and the Senior class 7 votes. Determine the power index for each class.

Total number of votes = 10 + 8 + 7 =25

25/2 = 12.5, so 13 votes are needed to pass an issue

Winning coalitions:

{So, Jr}, {So, Sr}, {Jr, Sr}, {So, Jr, Sr}

So: required in the first two winning coalitions, so PI = 2

Jr: required in the 1st and 3rd winning coalitions, so PI = 2

Sr: required in the 2nd and 3rd winning coalitions, so PI = 2

  1. Consider a situation where four individuals have weighted votes: A has 20 votes; B has 15 votes; C has 10 votes; and D has 5 votes. Determine the power index for each person.

Total number of votes = 20 + 15 + 10 + 5 = 50

50/2 = 25, so 26 votes are needed for a majority

Winning coalitions:

{A, B}, {A, C}

{A, B, C}, {A, B, D}, {A, C, D}, {B, C, D}

{A, B, C, D}

A: required in winning coalitions 1, 2, 3, 4, and 5: PI = 5

B: required in winning coalitions 1, 4, and 6: PI = 3

C: required in winning coalitions 2, 5, and 6: PI = 3

D: required in winning coalition 6: PI = 1