Math 110: Methods I(b)

0. What is the difference between a spoiler candidate in an and by candidates?

1. a) Consider the following preference results with for an election with four choices. Who is the winner? Does this person have a majority of the votes? Is plurality (the system used in the U.S.) give a satisfactory outcome in this case?

6 Voters 6 Voters 7 Voters 2 Voters 2 Voters Plurality Plurality w/IRV Vote-for-Two A 1 4 4 1 3 B 4 1 2 3 2 C 2 2 3 2 1 D 3 3 1 4 4

b) Who is the plurality with instant runoff winner (Plurality w/IRV)?

The Plurality w/IRV Method. The two candidates with the most first-place votes are pitted against each other. With preference , the first-place votes are re-assigned to the candidate with the higher .

c) Who is the vote-for-two winner?

The Vote-for-two Method. Each voter votes for two candidates. The candidate with the most votes wins. On a preference ballot, we assume the two candidates are those ranked 1 and 2.

2. The Hare method, also called (STV), is another type of plurality method with a different runoff process. This method eliminates one candidate at a time rather than just having an instant runoff between the top two candidates. The Hare method is used in .

The Hare Method. Each voter ranks the candidates in order of preference. The single candidate with the least number of first-place votes is eliminated and re-ranking takes place as needed. This elimination process continues until a candidate has a majority of first-place votes.

Using the below, who is the Hare winner?

6 Voters 6 Voters 7 Voters 2 Voters 2 Voters Round 1 Round 2 Round 3 A 1 4 4 1 3 B 4 1 2 3 2 C 2 2 3 2 1 D 3 3 1 4 4 3. The is another type of plurality method. The name comes from the 18th century mathematician and philosopher .

Condorcet Winners: Head-to-Head Comparisons. Another method of determining the winner of an election when we know the preference of each voter involves pitting each candidate against every other candidate in a series of head-to-head comparisons. If one candidate defeats (is defeated by) all others head-to-head, that candidate is the Condorcet Winner (Loser). There need not be a Condorcet winner or loser.

a) Determine the winners of all the head-to-head contests.

6 Voters 6 Voters 7 Voters 2 Voters 2 Voters A vs B A vs C A vs D B vs C B vs D C vs D A 1 4 4 1 3 B 4 1 2 3 2 C 2 2 3 2 1 D 3 3 1 4 4

b) Is there a Condorcet winner? Explain.

c) Is there a Condorcet loser? Explain.

d) A Copeland winner (loser) is the candidate who wins (loses) the most head-to-head . Is there Copeland winner or loser? Explain.

4. The is voting method that uses all of the preference information in a single stage. Jean-Charles de Borda devised the system in 1770. He was a French mathematician and political scientist. In the Borda Count, each candidate (or alternative) gets 1 point for each first-place vote received, 2 points for each second-place vote, etc., all the way up to N points for each last-place vote (where N is the number of candidates/alternatives). The candidate with the smallest point total is the Borda winner of the election.

a) Who is the Borda count winner?

6 Voters 6 Voters 7 Voters 2 Voters 2 Voters Borda Count Approval A 1 4 4 1 3 B 4 1 2 3 2 C 2 2 3 2 1 D 3 3 1 4 4

b) . Voters checkoff every candidate that they find satisfactory in an election. The candidate with the most checks wins. Suppose in this election everyone ranked 1 or 2 is ‘approved’ by the voters. (Insert checkmarks accordingly.) Who is the approval winner?

5. Compare the results of all the election methods. Is there a method that you believe is most fair or seems to best represent the voter’s sentiment? Why? Math 110: Group Hand In. Names:

Show your work or explain.

6. a) The preference schedule below represents an election among George (G), Holly (H), James (J), and Inez (I). Who is the plurality winner? Does the person have a majority of the votes?

13 Voters 12 Voters 10 Voters

G 1X 4 2X H 2X 2X 3 I 4 1X 4 J 3X 3 1X

b) Who is the plurality with instant runoff winner? Show your work making an extra column(s) in the table.

c) Who is the vote-for-two winner?

d) Who is the Borda count winner?

e) Who is the Hare winner? Show your work making an extra column(s) in the table below.

13 Voters 12 Voters 10 Voters

G 1X 4 2X H 2X 2X 3 I 4 1X 4 J 3X 3 1X

f) Who is the approval winner? (Give the vote counts.)

g) Is there a Condorcet winner? Loser? (Show the individual comparisons.)

h) Who is the Copeland winner? Loser?

Over 7. a) Extra Credit. Construct an election with four candidates A, B, C and D that has no majority winner. However, A is both the plurality winner and a Condorcet loser, B is the instant runoff winner and beats A head-to-head, and C is the Condorcet winner but is never ranked first by voters. Use 9 total votes arranged as follows. Create the voter preference rankings.

4 Voters 3 Voters 2 Voters A B C D

b) Verify that A is both the plurality winner and a Condorcet loser by using the appropriate vote counts. (Add and label additional columns to the table if helpful.)

c) Verify that B is the instant runoff winner and beats A head-to-head.

d) Who is the Borda count winner?

e) Who is the Hare winner?

f) Suppose in this case everyone ranked 1, 2, or 3 is ‘approved’ by the voters. Who is the approval winner?

8. Still have time? Work on Mindscape 1.9 on page 11 of the orange handout. It will be assigned next week.