1. MAT 105 Fall 2008
More Voting Methods

2. More Methods
There are many more methods for determining the winner of an election with more than two candidates
We will only discuss a few more:
sequential pairwise voting
Hare system
plurality runoff

3. Sequential Pairwise Voting
Idea: We like pairwise voting (where we can use majority rule), but if we look at all pairwise elections, we sometimes don’t get a winner
In sequential pairwise voting, we put the candidates in order on a list, called the agenda

4. How It Works
We pit the first two candidates on the agenda against each other. The winner moves on to face the next candidate on the list, and so on. The candidate remaining at the end is the winner.
This process resembles a tournament bracket, and has the advantage that, unlike Condorcet’s method, we always get a winner

5. An Example
Let’s use sequential pairwise voting with this profile and the agenda A, B, C, D
Voters
Preference Order 4
A > B > D > C 3
C > A > B > D
B > D > C > A A
A beats B, 7-3 A
C beats A, 6-4 B C
D beats C, 7-3 C D D

6. Problems
If we look closely at this agenda, we notice that every single voter prefers B over D, and yet D was our winner!
In fact, by cleverly choosing the right agenda, we could make any of the four candidates win this election
Sequential pairwise voting does not satisfy the Pareto condition
Voters
Preference Order 4
A > B > D > C 3
C > A > B > D
B > D > C > A

7. The Pareto Condition
If every voter prefers one candidate over another, then the latter candidate should not be among the winners of the election
Named for Vilfredo Pareto ( ), Italian economist
Does plurality satisfy the Pareto condition?

8. Another Method: The Hare System
Also known as Instant Runoff Voting, this system is used for various elections in the US, Canada, the UK, Ireland, and Australia
Repeatedly delete candidates that are “least preferred” in the sense of being at the top of the fewest ballots. If there is a tie, eliminate all of the tied candidates, until there is no one left to eliminate

9. An Example
Voters
Preference Order 5
A > B > C 4
C > B > A 3
B > C > A 2
B > A > C
In this example, A has 5 first-place votes, B has 5 first-place votes, and C has 4 first-place votes, so C is eliminated
Now A has 5 first-place votes, and B has 9, so A is eliminated
B is the only candidate left, so B is the winner
Voters
Preference Order 5
A > B 4
B > A 3 2

10. Another Example
Voters
Preference Order 5
A > B > C 4
C > B > A 3
B > C > A 1
B > A > C
This time, A has 5 first-place votes, and B and C are tied with 4, so B and C are both eliminated at the same time
This leaves only A to win the election

11. Problems
Now let’s modify the profile from the previous example, so that the 1 voter with preference B > A > C now has preference A > B > C
Notice that this change moves the winner higher on that voter’s ballot
Voters
Preference 6
A > B > C 4
C > B > A 3
B > C > A
Voters
Preference 6
A > C 4
C > A 3
C wins!

12. Why is this a problem?
A was the winner of the original election, and one of the voters changed his ballot to move A higher, causing A to lose
This shows that the Hare system is not monotone

13. Monotone
A voting system is monotone if whenever a candidate is a winner, and a new election is held where the only change is for some voter to move that winner higher on his/her ballot, then the original winner should remain the winner
The Hare system is not monotone, but despite this drawback it is one of the more common alternative voting systems in use today

14. One More Method: Plurality Runoff
Hold a plurality election, but if no candidate receives a majority, we hold a runoff election
The runoff election is between the two candidates who received the most first-place votes in the original election
In case of ties, there might be more than two candidates with the most first-place votes, so we use plurality to decide a winner between those candidates only

15. Example
Voters
Preference Order 5
A > B > C 4
C > B > A 3
B > C > A 2
B > A > C
In this profile, A gets 5 first-place votes, B gets 5 first-place votes, and C only gets 4
The runoff is between A and B
B wins the runoff 9 votes to 5

16. Another Example
Voters
Preference Order 4
A > B > C > D 3
C > D > B > A
B > C > D > A 2
D > B > A > C
In this profile, A has 4 first-place votes, B has 3, C has 3, and D has 2
The runoff is between A, B, and C
We use plurality to decide the winner; keep in mind that the 2 voters who like D best get to vote in the runoff!
B wins the runoff with 5 votes